A novel approach towards a lubricant-free deep drawing

A novel approach towards a lubricant-free deep

drawing process via macro-structured tools

Von Der Fakultät Maschinenwesen

der

Technische Universität Dresden

zur

Erlangung des akademischen Grades

Doktoringenieur (Dr.-Ing.)

angenommene Dissertation

Ali Mousavi, M.Sc.

Geboren am 18.09.1985 in Iran

Tag der Einreichung: 24.05.2019

Tag der Verteidigung: 17.10.2019

Gutachter: Prof. Dr.-Ing. Alexander Brosius

Prof. Dr.-Ing. habil. Marion Merklein

Vorsitzender der Promotionskommission: Prof. Dr.-Ing. habil. Thorsten Schmidt

„If you are not part of the solution, you are part of the problem“

ELDRIDGE CLEAVER

Acknowledgement

This thesis is written during my times as a research assistant at the chair of Forming and

Machining Processes at TU Dresden.

Firstly, I would like to express my sincere gratitude to my advisor Prof. ALEXANDER BROSIUS

for the continuous support of my Ph.D study and related research, for his patience, motivation,

and immense knowledge. His guidance helped me in all the time of research and writing of

this thesis. I have been extremely lucky to have a supervisor who cared so much about my

work, and who responded to my questions and queries so promptly.

Besides my advisor, I would like to thank the rest of my thesis committee: Prof. MARION

MERKLEIN, Prof. ANDRÉS LASAGNI, Prof. MICHAEL SCHOMÄCKER and Prof. UWE FÜSSEL for their

insightful comments and encouragement.

I would also like to thank all staff members at chair of Forming and Machining Processes

especially my closest colleagues CHRISTINA GUILLEAUME, NIKLAS KÜSTERS, RÉMI LAFARGE,

CHRISTIAN STEINFELDER, MARC TULKE and ALEXANDER WOLF for the stimulating discussions, for

the sleepless nights we were working together before deadlines, and for all the fun we have

had in the last years.

I would also like to extend my thanks to the technicians of the laboratory of the department

for their help in preparation of equipment for the experiments.

I am most grateful to my family members. My parents have always loved me and supported

my every choice. As I know, they are the happiest and the most proud when seeing their son

gets this degree, I dedicate this thesis to them.

Dresden, October 2019 ALI MOUSAVI

Abstract

In today’s industry, the sustainable use of raw materials and the development of new green

technology in mass production, with the aim of saving resources, energy and production costs,

is a significant challenge. Deep drawing as a widely used industrial sheet metal forming

process for the production of automotive parts belongs to one of the most energy-efficient

production techniques. However, one disadvantage of deep drawing regarding the realisation

of green technology is the use of lubricants in this process. Therefore, a novel approach for

modifying the conventional deep drawing process to achieve a lubricant-free deep drawing

process is introduced within this thesis.

In order to decrease the amount of frictional force for a given friction coefficient, the integral

of the contact pressure over the contact area has to be reduced. To achieve that, the flange

area of the tool is macro-structured, which has only line contacts. To avoid the wrinkling, the

geometrical moment of inertia of the sheet should be increased by the alternating bending

mechanism of the material in the flange area through immersing the blankholder slightly into

the drawing die.

Therefore, the developed process can, besides the reduction of the contact area and the

blankholder force, also increase the resistance of the sheet metal against wrinkling.

Furthermore, through adjusting the immersion depth, it is possible to control the material flow

and enlarge the process window. The induced alternating bending to stabilise the sheet metal

against wrinkling during deep drawing with macro-structured tools generates an additional

force in the drawing direction. The created non-frictional restraining force can be used

positively to compensate the springback behaviour of the workpiece.

Table of contents

i

Table of contents

I. Symbols and abbreviations............................................................................................. iv

1. Introduction ..................................................................................................................... 1

2. State of the art ................................................................................................................ 3

2.1 Fundamentals of deep drawing process .................................................................. 3

2.1.1 Total forming force of a rotationally symmetric deep drawn cup ....................... 5

2.1.2 Process window in deep drawing ..................................................................... 6

2.2 Tribology in metal forming ....................................................................................... 7

2.2.1 Lubrication mechanisms in metal forming ........................................................ 7

2.2.2 Friction models in metal forming ...................................................................... 9

2.2.3 Diversity of lubricants in metal forming ........................................................... 12

2.3 Economic and ecological disadvantages of lubrication ........................................... 13

2.4 Effects of lubricants on the quality of the deep drawing process ........................... 14

2.5 Possibilities for friction reduction in the deep drawing process.............................. 15

2.5.1 Surface coating ............................................................................................... 16

2.5.2 Surface micro-structuring................................................................................ 19

2.6 Tribological testing methods .................................................................................. 22

2.6.1 Tribometer ...................................................................................................... 22

2.6.2 Strip-drawing and the draw-bend test ............................................................. 23

2.7 Summary of Chapter 2 ........................................................................................... 24

3. Objectives ..................................................................................................................... 26

4. Methodology and approach ........................................................................................... 28

4.1 Macro-structuring: Tip to Tip .................................................................................. 30

4.2 Macro-structuring: Tip to Hutch ............................................................................. 30

Table of contents

ii


5. Experimental setup ....................................................................................................... 35

5.1 Press machines ..................................................................................................... 35

5.1.1 Hydraulic press BUP 600 ................................................................................ 35

5.1.2 Hydraulic press Röcher RZP 250..................................................................... 36

5.2 Tooling material ..................................................................................................... 37

5.3 Workpiece materials .............................................................................................. 37

5.4 Conventional and macro-structured tools ............................................................... 40

5.4.1 Tools for draw bending of U-Channels ............................................................ 40

5.4.2 Tools for deep drawing of axial symmetric parts ............................................. 41

6. Process modeling and determination of process criteria ............................................... 43

6.1 Process modelling: buckling stability...................................................................... 44

6.1.1 Identification of the most unstable part of the flange area .............................. 44

6.1.2 Buckling analysis of the free end part of the sheet ......................................... 45

6.2 Process modelling: forming energy ....................................................................... 46

6.3 Extension of the model for deep drawing of non-rotationally symmetric parts ....... 50

6.4 Development of a criterion for the prediction of wrinkling ..................................... 51

6.5 Development of a criterion for prediction of bottom cracks ................................... 54

6.6 Developement of a criterion for prediction of dimensional accuracy of the parts ... 56


7. Results .......................................................................................................................... 61

7.1 Investigation about the effects of macro-structuring on process characteristics .... 61

7.1.1 Investigation about friction reduction through macro-structuring .................... 61

7.1.2 Investigation about process stabilisation through macro-structuring ............... 63

Table of contents

iii

7.1.3 Investigation about springback reduction through macro-structuring .............. 66

7.2 Verification of the developed analytical methods ................................................... 69

7.2.1 Verification of the developed process model: forming energy ........................ 69

7.2.2 Verification of the developed criterion: prediction of wrinkling ........................ 71

7.2.3 Verification of the developed criterion: prediction of bottom cracks ................ 72

7.2.4 Verification of the developed criterion: prediction of dimensional accuracy .... 73


8. Transfer of the methodology into the complex geometries ........................................... 77

8.1 Construction of a deep drawing tool to form a T-Cup ............................................. 77

8.2 Determination and evaluation of optimum initial blank geometry ........................... 79

8.3 Test Procedure and interpretation of results .......................................................... 81


9. Possibilities for process improvement .......................................................................... 87

9.1 Use of rotary elements as macro-structured elements .......................................... 87

9.2 Macro-structured tools with variable wavelengths ................................................. 88


10. Summary and conclusions ............................................................................................ 92

II. Zusammenfassung ....................................................................................................... 95

III. List of Figures ............................................................................................................... 98

IV. List of Tables .............................................................................................................. 103

V. References ................................................................................................................. 104

Symbols and abbreviations

iv

I. SYMBOLS AND ABBREVIATIONS

Symbol Unit Meaning

a mm Length of rectangular element in buckling analysis

a - Material constant

A0 mm2 Cross sectional area of the drawn part

A0 mm2 Original area of the sheet metal before forming

A80 % Elongation at fracture

A1 mm2 Increased area after stretch drawing

Aa mm2 Apparent contact area

Ai mm2 Partial contact area in macro-structured tool

Ar mm2 Real contact area

b mm Width of rectangular element in buckling analysis

b mm Flange width

CR - Crack factor

E GPa YOUNG modulus

E0 GPa Plastic buckling modulus

EB Nm Bending energy into a half sine wave segment in the wrinkled flange

Eb Nm Bending energy

Ef Nm Frictional energy

Eid Nm Ideal forming energy

EL Nm Energy due to the lateral loading of the surface in the wrinkled flange

Et Nm Energy due to the tangential compressive forces in the wrinkled flange

f - Modified friction factor

F1 kN Pulling force

F2 kN Back tension force

Fb kN Bending force

Fbc kN Bottom crack force

FBH kN Blankholder force

Ff kN Friction force

Ffd kN Friction force in die edge radius

Fff kN Friction force in flange area

Fid kN Ideal forming force

FN kN Normal force

Ftot kN Total punch force

h mm Punch displacement / drawing depth

I kg m2 Moment of inertia

k MPa Shear yield stress


v

l mm Length of a half sine wave segment

m - Friction factor

m - Characteristic strain constant

m - Number of half sine waves in tangential direction in the buckled mode

Mb Nm Bending moment due to tangential stresses over the sheet thickness

𝑀beb Nm Required elastic bending moment to close the split ring

𝑀bres Nm Bending moment because of residual stress in the intact ring

n - Flattering factor for the intermediate transition from the COULOMB to

the shear friction model

n - Number of half sine waves in radial direction in the buckled mode

n - Strain hardening coefficient

P MPa Surface pressure

P - Slope of the stress–strain curve at a given value of strain

Pi MPa Partial surface pressure in macro-structured tool

PTotal MPa Total surface pressure in macro-structured tool

q mm2 Flange surface

r mm Radius of an arbitrary point in flange area

�̅� - Mean vertical anisotropy

r0 mm Blank initial radius

r1 mm Mean radius of cut out ring

r2 mm Mean radius of cut out and split ring

ra mm Blank current outer radius

Ra μm Arithmetical mean deviation of surface roughness

rb mm Bending radius

ri mm Inner radius of the die

ri* mm Inner radius of the flange

RC mm Corner radius

RM mm Die edge radius

Rm MPa Ultimate tensile strength

rm mm Mean flange radius

Rmax μm Maximum depth of the assessed profile of surface roughness

rp mm Punch edge radius

rs mm Radius of macro-structures

rθ mm Radius of the die edge from the centre

Rz μm Average distance between the highest peak and lowest valley of

surface roughness

s0 mm Sheet thickness

t s Time

v mm.s-1 Sliding velocity between two friction bodies


vi

V mm3 Volume

w mm Deflection of plate in z-direction in buckled mode

α - Contact area ratio

α Rad Coordinate of the local intersection coordinate system

α Rad Deflection angle of punch edge radius

β - Drawing ratio

β1 Rad Springback angle regarding the sidewall of the U-Channel

β2 Rad Springback angle regarding the flange of the U-Channel

δ mm Immersion depth

Δ mm Opening gap of split ring

φt - Tangential compression factor at the outlet of punch edge radius

δh mm Virtual punch displacement

δrθ mm Virtual displacement at outlet of the die edge radius

δu mm Virtual displacement at the flange inner edge

δV mm3 Virtual volume change

δWext Nm Virtual external work

δWint Nm Virtual internal work

δεij - Virtual strain

εpl - Plastic strain

휀max,outeb -

Maximum of tangential elastic bending strain due to the springback at

the outer surface of the intact ring

휀max,inneb -

Maximum of tangential elastic bending strain due to the springback at

the inner surface of the intact ring

η - Triaxiality

η Pa.s Lubricant viscosity

ηF - Forming efficiency

θ Rad Bending angle

Λ μm Period of micro-structures

λ mm Wavelength of macro-structures

μ - Friction coefficient

ν - Poisson's ratio

ρ mm-1 Curvature in the sidewall of the U-Channel

Σ MPa External stress

σ∞ MPa Asymptotic value for saturation strength

σeq MPa Equivalent stress

σI MPa First principle stress

σII MPa Second principle stress

σij MPa Internal stress


vii

σIso MPa Pure isotropic stress

σKin MPa Pure kinematic stress

σm MPa Mean stress

σn MPa Normal stress

σr MPa Radial stress

σt MPa Tangential stress

σt,cr MPa Critical tangential stress

σy MPa Yield stress

σy,1 MPa Yield stress before bending over the die edge radius

σy,2 MPa Yield stress after bending over the die edge radius

σy0 MPa Initial yield stress

σym MPa Mean yield stress

σym,1 MPa Average yield stress over the flange area

σym,2 MPa Average yield stress over the die edge radius

𝜎maxeb MPa Maximum of tangential elastic bending stress over the sheet thickness

𝜎max,innel MPa

Maximum of tangential elastic bending stress at the inner surface of

the cup

𝜎max,outel MPa

Maximum of tangential elastic bending stress at the outer surface of

the cup

τa MPa Average shear stress at contacting asperity peaks of surface

roughness

τb MPa Average shear stress in the valleys of surface roughness

τf MPa Frictional shear stress

BUP Blech Universal Prüfmaschine (English: sheet metal universal testing

machine)

CVD Chemical Vapour Deposition

DLC Diamond Like Carbon

DLIP Direct Laser Interference Pattering

EDM Electrical Discharge Machining

FEM Finite Element Method

o/w Oil in Water

PVD Physical Vapour Deposition

ta-C Tetrahedral amorphous carbon

Introduction

1

1. INTRODUCTION

Metal forming belongs to one of the oldest technologies. During the first industrial revolution

(18th -19th century), a mostly experience-based industrial metal forming technology developed

for the first time after centuries or even millennia of traditional craftsmanship. It matured

during two subsequent phases of development from approximately 1920 to 1960 with the

establishment of the first scientific background of plasticity theory, materials technology, and

experimental process analysis. Since 1960, the third phase of metal forming technology

development was characterised by the introduction of the computer that revolutionised nearly

all aspects of metal forming: process analysis and optimised design, materials technology and

science, process-oriented tool technology, metrology and process control, etc. Consequently,

product quality, productivity, flexibility, and economy were substantially improved [1]. Among

the metal forming techniques, sheet metal forming is one of the most frequently used primary

methods to produce different variety of components. Figure 1-1 shows for example the

majority of the formed sheet metal parts in a car body.

Figure 1-1: Structure of Audi A4 limousine with hang on parts [2]

Deep drawing as a widely used industrial sheet metal forming process for the production of

automotive parts, households, etc. belongs to one of the most energy-efficient production

techniques, based on its high material utilisation. In today’s industry, the sustainable use of

raw materials and the development of new green technology in mass production, with the

Cold-formed steel

Hot-formed steel

Cast aluminium

Aluminium sections

Deep drawn aluminium

Introduction

2

aim of saving resources, energy and production costs, is a significant challenge. But one

disadvantage of deep drawing regarding the realisation of green technology is the use of

lubricants in this process. Therefore, a novel approach for modifying the conventional deep

drawing process to achieve a lubricant-free deep drawing process is introduced within this

thesis.

Generally, in deep drawing the process window is limited by the occurrence of wrinkles and

bottom cracks. Increasing the friction in the deep drawing process leads to an increase of the

total punch force and as a result the risk of bottom cracks and a decrease of the process

window become more probable. Moreover, in most cases, increasing the friction between

the tool and the workpiece reduces the lifetime of the tool. That is why lubricants are mostly

inevitable in the metal forming, especially in deep drawing for successful operation and

reduction in energy consumption. However, the huge amount of lubricant used has economic

and ecological disadvantages in today’s industry. Regarding economic aspects, using mineral-

oil-based lubricants leads to an increase of production steps, because an additional post-

treatment process is required for cleaning the workpiece by means of degreasing agents

which are usually solvent-based [3]. Moreover, more than 15% of the total cost of sheet metal

forming is spent on lubricating liquids, including buying the lubricants, machines for their

application and cleaning lubricated parts [4]. Besides that, lubricants are often either harmful

to health or harmful to the environment. Therefore, it is of great interest to develop a process

which permits a significant reduction of the amount of lubricating agents or even lubricant-

free metal forming. However, this poses a great difficulty in application for sheet metal

forming in general. In recent years, numerous studies on the application of dry lubricants, as

well as environmentally friendly lubricants during the forming process, have been performed

[5]. Various tool materials and coatings have superior tribo-properties and have been

investigated widely for general purposes in tribology. Nevertheless, all of these studies and

proposed methods are based on laboratory conditions and none of them can realise a total

lubricant-free forming process [6]. Therefore, a process without any lubrication with the

capability to transfer into industrial application is of major interest. Within the scope of this

thesis, a new, lubricant-free, deep drawing process is developed, which ensures the process

window despite the absence of lubricants. Since the largest contribution of frictional force in

the deep drawing process is on the die edge radius and flange area, these parts of the tool

should be adapted for a stable lubricant-free deep drawing process. Reduction of friction on

the die edge radius through tool coating and micro-structuring is already state of the art.

However, a new tool design should be developed to reduce the amount of friction in the flange

area of the drawing tool. Therefore, in order to decrease the present friction forces for a given

friction coefficient, the integral of contact pressure over the whole contact area should be

reduced. Thus, as one of the most promising methods, macro-structured deep drawing tools

have been developed within the scope of this thesis to reduce the contact area with contact

lines or points. For a time efficient process design and also to analyse the influence of process

parameters on the process stability in advance, the newly developed process was modelled

analytically. The results of the model were verified through experimental tests as well as FEM.

State of the art

3

2. STATE OF THE ART

In this chapter, the fundamentals of the deep drawing process are introduced. Furthermore,

the advantages and disadvantages of using lubricants for a stable and efficient process are

determined. Moreover, the already existing approaches for the reduction of friction in the deep

drawing process are studied, with the aim of avoiding lubricants and to reach the vision of a

lubricant-free deep drawing process.

2.1 FUNDAMENTALS OF DEEP DRAWING PROCESS

Nowadays, the classification of the forming processes can take place with regard to the

temperature (cold forming, warm forming, or hot forming), process technology (sheet and bulk

metal forming), and stress state [7]. According to DIN 8582 [8], depending on the main

direction of the applied stress, they can be subdivided into five groups:

Forming under compressive conditions,

Forming under combined tensile and compressive conditions,

Forming under tensile conditions,

Forming by bending,

Forming under shear conditions.

In order to assess the relative economic importance of sheet metal forming compared with

other procedures, Table 2-1 gives an overview.

Table 2-1: Comparison between different forming processes regarding their economic importance [9]

Process Sheet metal

forming Hot forming Cold forming Warm forming

Relative ratio

of production 100 10 1 0.2 – 0.3

DIN8584-3 defines the deep drawing process as follows: “Deep drawing is the forming of a

flat sheet blank (or foil/plate, section/blank, depending on the material) into a hollow shape or

of a hollow shape to a hollow shape with smaller perimeter by pressing it through a die without

intended change of the blank thickness” [10]. Its main fields of application are now in the

automotive industry (structural and body components), in the branch of components made by

sheet metal for household applications (dishwashers, washing machines, catering containers,

etc.), as well as in the aerospace sectors [11].

Deep drawing of sheet metal is performed with a punch, blankholder, and drawing die.

Generally, a basic deep drawing operation can be the forming of a sheet blank into a

rotationally symmetric cup as illustrated in Figure 2-1. Understanding the material flow during

the process is essential for understanding the present stresses on the workpiece. Considering

State of the art

4

a rotationally symmetric deep drawing process, the material under the punch is forced into

the cavity, pulling material in the flange area into the hole. Because of the applied tensile

stress in the radial direction, the outer radii in the flange area decrease continuously, and this

leads to induce a compressive stress in the tangential direction. It can cause an uneven

distribution of thickness in the flange area, with the minimum at the inner and maximum at

the outer part. Unless holding down pressure is applied from the blankholder, the induced

compressive stress will cause the blank to buckle. In order to prevent the buckling of the blank

and be able to control the material flow, the blankholder with predefined blankholder force FBH

press the blank in the normal direction. When the blank passes the flange area, it is subjected

to bend and slide over the die edge radius. After bending, the blank unbends to flow

downward along the part wall. However, as the complexity goes up, the manufacturing

difficulties increase. Deep drawing for complex geometries and large height-diameter ratio

components is generally done in multiple steps due to limitations regarding the properties of

the sheet material, especially its formability [12]. The most commonly used sheet materials

for deep drawing applications are steel and aluminium alloys because of their mechanical

properties, natural availability, and cost-effectiveness. Among them, cold-rolled steels

because of their good ductility [13] and aluminium-magnesium alloys due to their excellent

balance between formability and strength [14] are usually used for deep drawing. The tool is

subjected to repeated contacts with the blank material during forming processes.

Figure 2-1: Deep drawing mechanism, geometrical variables, and acting stresses [10]

For better understanding the process mechanism and also to find out the interrelationship

between the process-relevant parameters, it is required to analyse the total punch force

regarding its constituent parts. In the following, the parameters affecting the punch force are

introduced.

Normal stress

Tangential stress

Radial stress

Friction stress

:

:

:

:

• Flange area

• Main forming area

• Tensile-compression stress

• Part wall area

• Plain strain

• Side fixed uniaxial tension

• Bottom area

• Stretch forming

• Biaxial tension

FBH : Blank holder force

r0 : Sheet initial radius

ra : Sheet current radius

ri : Die inner radius

s0 : Sheet thickness

FBHFBH

ra

r0

riDie

Sheet metal

Blankholder

s0

s0

State of the art

5

2.1.1 Total forming force of a rotationally symmetric deep drawn cup

Considering the rotationally symmetric deep drawing process, the total forming force Ftot

consists of the superposition of four individual forces as follows:

𝐹tot = 𝐹id + 𝐹b + 𝐹ff + 𝐹fd 2-1

Here, Fid is the ideal forming force in the flange area, Fb is the bending force on the die edge

radius, Fff are the friction forces between the workpiece and the tool in the flange area, and

Ffd is the frictional force between the workpiece and the die edge radius. These forces can be

calculated individually based on SIEBEL`s calculations [15] as follows:

𝐹id = 2 ∙ 𝜋 ∙ 𝑟i ∙ 𝑠0 ∙ 1.1 ∙ 𝜎ym,1 ∙ ln𝑟0𝑟i

2-2

𝐹b = 2 ∙ 𝜋 ∙ 𝑟i ∙ 𝑠02 ∙𝜎ym,2

4 ∙ 𝑅M 2-3

𝐹ff = 2 ∙ 𝜇 ∙ 𝐹BH 2-4

𝐹fd = (𝑒𝜇∙𝛼 − 1) ∙ (𝐹id + 𝐹ff) 2-5

In these equations, r0 and ri designate the initial radius of the sheet and the cup radius

respectively, σym,1 is the average yield stress over the flange area, RM denotes the bending

radius at the die edge radius, σym,2 is the average yield stress over the die edge radius, FBH is

the blankholder force, μ is the COULOMB friction coefficient and α is the deflection angle at the

die edge. Figure 2-2 represents a schematic overview of these force components.

Figure 2-2: Individual components of the total forming force in deep drawing of the rotationally

symmetric cup

The total deep drawing punch force is the sum of all individual force components:

σt

σr+dσr

r

dα

dr

ri

s0

RM

Ideal forming force Bending force on die

edge radius

Friction force on the

flange area

Friction force on die

edge radius

ra

State of the art

6

𝐹tot = 2 ∙ 𝜋 ∙ 𝑟i ∙ 𝑠0 ∙ [𝑒𝜇∙𝛼 (1.1 ∙ 𝜎ym,1 ∙ ln (

𝑟0𝑟i) +

𝜇 ∙ 𝐹BH𝜋 ∙ 𝑟0

) + 𝜎ym,2 ∙𝑠0

2 ∙ 𝑅M] 2-6

As the Equation 2-6 implies, there are many parameters, which influence the punch force, like

the geometry of the workpiece, sheet thickness, material properties, friction coefficient and

blankholder force. For a stable process, a good balance is required between the process

parameters. In general, the working area for a stable deep drawing can be described through

the process window.

2.1.2 Process window in deep drawing

In the deep drawing process, wrinkling, bottom cracking and earing are the most common

process defects. Control of material flow is one of the most important issues in the deep

drawing process for preventing defects in the deep drawn part [16]. Earing is one of the

characteristic defects observed during the deep drawing process due to the anisotropic nature

of sheet metal, which is defined as the formation of waviness on the uppermost portion of

the deep drawn cup and can be reduced by modifying the initial blank geometry [17]. Plastic

buckling in the flange area of the deep drawing process occurs because of high compressive

stress and inadequate blankholder force leading to wrinkling. In order to ensure a stable

process regarding the wrinkling, a high amount of blankholder force is required [18]. However,

it increases the normal surface pressure and results in an increase of frictional forces in the

flange area. Increasing the friction in deep drawing due to the relative motion between tools

and the workpiece in the flange area, as well as the die edge radius under a high surface

pressure, leads to an increase of total punch force which consequently results in exceeding

the formability limits of the material. As a consequence, the risk of bottom cracking becomes

more probable. Bottom cracks indicate material instability caused by strain localisation, when

the plastic deformation carried out by the punch force exceeds the formability limit of the

material [19]. Wrinkling and bottom cracking can lead to an immediate loss of part properties

and functionality. Therefore, in deep drawing, the process window, which can be

characterised as the working area for the production of faultless parts by means of a stable

process, is limited by the occurrence of wrinkles and bottom cracks (see Figure 2-3).

Figure 2-3: Process window in deep drawing

Bla

nkhold

er

forc

eF

BH

• Process window

• Faultless parts

• Stable process

Drawing depth h

Bottom cracks

Friction

Wrinkles

State of the art

7

Generally, achieving the maximum process window and increasing the product quality are the

indisputable targets in industrial production. That is why the control of material flow in deep

drawing for enlargement of the process window is of importance. However, there are some

methods to control the material flow in order to reach the greatest possible process window.

For example, the geometry of the sheet metal plays an important role in material flow.

Therefore, for deep drawing of complex geometries, the sheet metal geometry should be

optimised to reach the greatest process window [20]. Furthermore, drawbeads can also be

used to control the material flow during the process. Drawbeads restrain the material flow,

causing a change of the strain distribution, which consequently hinders the material flow [21].

Above all, the reduction of friction can postpone the occurrence of bottom cracks and

consequently the enlargement of the process window [22]. Hence, besides an optimum tool

design, an excellent tribological system in the deep drawing process is essential for increasing

the process quality and the enlargement of the process window through prevention of process

failures.

2.2 TRIBOLOGY IN METAL FORMING

In the previous section, the mechanism of the deep drawing process was introduced and it

was pointed out that the tribological behaviour of the deep drawing process is of importance

to ensure a stable process with a large process window. The word tribology was first used in

England in the 1960's, and is defined as the science and technology of interacting surfaces in

relative motion. It comes from the Greek word “tribos” meaning to rub [23]. The term was

coined as a conscious attempt to combine the historically independent fields of friction,

lubrication and wear in an interdisciplinary manner [24]. Based on the lubricant viscosity,

entrainment velocity and the surface pressure, different lubrication mechanisms can take

place in the forming process. These mechanisms are discussed in the following section.

2.2.1 Lubrication mechanisms in metal forming

In sheet metal forming processes, the lubricant acts under a high normal pressure, and

therefore the mechanism of lubrication depends on the lubricant viscosity η and the process

properties (sliding velocity v and normal pressure P). The influence of these parameters can

be described by the Stribeck curve [25], in which the friction coefficient is controlled by a

single parameter produced by all three decisive factors. In a Stribeck curve (see Figure 2-4),

these parameters are plotted in a logarithmic diagram and present four primary lubrication

mechanisms (i.e. dry / boundary / mixed film / hydrodynamic) [26]. A dry (lubricant-free)

condition means no lubrication is supplied at the mating surfaces; therefore, the friction is high

and all the force is guided through from one body to another by contact at the asperities.

Moreover, this can lead to pure intermetallic contact and cold welding (see section 2.4).

Boundary lubrication is defined as a condition where the solid surfaces are so close together

that the surface interaction between single or multi-molecular films of lubricants and the solid

asperities dominates the contact, and force can be partially transmitted through this molecular

State of the art

8

layer [27]. In the ideal situation, when such boundary layers are formed, and the underlying

surfaces subsequently start to slide along each other, molecules of the lubricant are sheared

off or pulled off the surfaces, and are then replaced by a new lubricant molecule. Boundary

lubrication is the most widely encountered lubrication condition in the deep drawing process

because of the high normal pressure and the minimum distance between the blank and tools

[28]. With increasing load, the lubricant film becomes thinner and thinner, but as long as there

is only one monolayer of lubricant present, friction remains relatively low. Besides boundary

lubrication, mixed-layer lubrication is also frequently encountered in the deep drawing process

[28].

Figure 2-4: The Stribeck curve showing the onset of various lubrication mechanisms [29]

In this case, the micro-peaks of the metal surface experience boundary lubrication conditions

and the micro-valleys of the metal surface become filled with the lubricant [30]. At some parts

of the surfaces, the load is carried by the boundary layer, while at other parts a full lubricant

film is built up [31]. The hydrodynamic lubrication mechanism occurs when the two surfaces

are fully separated by a lubricant film. In this case, the relative tangential displacements lead

to shear in the lubricant film only. The resisting force is now caused by this shear in the

lubricant. In actual hydrodynamic lubrication, the lubricant is dragged into a lubricant film by

the relative velocity of the surfaces and the shape of these surfaces [32]. This mechanism can

be rarely seen in metal forming [28].

Understanding the physical background of the frictional condition in metal forming processes

is important for process design. Furthermore, for simulation procedures, depending on the

process, it is important to apply the proper friction model. In the following section, the most

Fricti

on

coeff

icie

ntμ

Film

thic

kness

Dry

Boundary

Mixed

Hydrodynamic

Dry

μ > 0.3

Boundary lubrication

0.1 < μ < 0.3

Mixed-Layer lubrication

0.03 < μ < 0.1

Hydrodynamic lubrication

μ < 0.03

η Lubricant viscosity:

v Sliding velocity:

P Surface pressure:

μ Friction coefficient:

State of the art

9

relevant friction models in metal forming are introduced and their fields of application are

discussed in detail.

2.2.2 Friction models in metal forming

A quantitative description of friction in different conditions comes from the first investigation

of THERMISTIUS in 350 BC [33]. He found that the friction for sliding is greater than that for

rolling. This conclusion led to the investigation in modern terms that the static friction

coefficient is greater than the kinetic. It was observed in the 1500's by DA VINCI [34], retrieved

by AMONTONS in 1699 [35], verified by EULER in 1750 and COULOMB in 1781 that friction is

proportional to the load and independent of the sliding contact area [36]. Thus, the friction

coefficient is independent of velocity in the case of dry sliding [37]. In the theory of plasticity,

this is usually known as the COULOMB friction model in the form:

𝜏f = 𝜇 ∙ 𝑃 2-7

Where τf denotes the frictionally-induced shear stress, μ is the COULOMB’s friction coefficient

and P is the normal surface pressure. In many forming processes, the normal surface pressure

P can reach a multiple of the yield stress of the material. At this point, the workpiece can start

to deform plastically by shearing at the tool interface. Thus, the linear relationship between τf

and P, as described by Equation 2-7, may not be valid at high surface pressure levels because

the frictional shear stress τf cannot exceed the shear yield stress k of the deformed workpiece

material. Therefore, the friction coefficient is no longer meaningful when μ·P exceeds τf. Thus,

to avoid this limitation of COULOMB’s model, the shear friction model can alternatively be used

[38]. This model assumes that the frictional shear stress τf must be linked with the shear yield

stress k of the softer material with the following relation:

𝜏f = 𝑚 ∙ 𝑘 2-8

Whereby the proportionality factor m is the friction factor (0 ≤ 𝑚 ≤ 1) and k is the shear yield

stress. In this model, m equals to zero for no friction, and m equals to unity for a sticking

friction condition, which is the case where sliding at the interface is purely by shearing of the

base material (adhesion). The shear yield stress k depends on the yield stress of material σy,

and yield criterion. When the Tresca yield criterion is considered, the shear friction model can

be written as follows:

𝜏f = 𝑚 ∙𝜎y

2 2-9

Moreover, by using the V. MISES yield criterion, it can be formulated as follows:

𝜏f = 𝑚 ∙𝜎y

√3 2-10

State of the art

10

The maximum of friction force in the shear friction model is independent of the normal

pressure, but depends on the size of the contact area. This leads to an overestimation of

friction in the case of low surface pressure. By summarising these two friction models, it can

be concluded that the COULOMB’s friction model is well suited for friction modelling with low

surface pressures, but it overestimates the friction stress under high contact pressure. On the

other hand, the shear friction model is suitable for high surface pressure, but it overestimates

the friction stress under low contact pressure. Beyond that, the real surface pressure is

dependent on the part of the contact area, which is participating in the friction. Since the tool

is rigid, it cannot be deformed. The workpiece surface can also be plastically deformed, which

leads to the enlargement of the surface topography and as a result affects the real surface

pressure. These loading parameters, i.e. surface pressure and surface enlargement, can be

used in metal forming to describe the process and thus to select the tribological system and

the tool design. It is more relevant in bulk metal forming because of the relative high surface

pressure (locally even up to 3000 MPa) and the surface enlargement (in special cases up to

20) [39]. Figure 2-5 compares the range of surface pressure and the resulting surface

enlargement between the bulk and sheet metal forming process.

Figure 2-5: Comparison between bulk and sheet metal forming regarding the surface pressure and

the resulting surface enlargement according to [40]

To take the effect of surface pressure in the friction calculation into account, OROWAN

combined in Equation 2-11 both the friction models mentioned above (Equations 2-7 and 2-8),

together with the goal of using the advantages of COULOMB’s model for low surface pressure

and adequacy of the shear friction model for high surface pressure [41].

𝜏f =

{

𝜇 ∙ 𝑃 , 𝑃 ≤𝑚 ∙ 𝑘

𝜇

𝑚 ∙ 𝑘 , 𝑃 >𝑚 ∙ 𝑘

𝜇

2-11

However, a disadvantage of the Orowan friction law is the sudden transition from the

COULOMB to the shear friction model. Since the first slipping of the surfaces begins locally in

small regions, this is not justified physically with the OROWAN model. In addition, the sudden

change of the friction model has a negative effect on the stability of a numerical

implementation. SHAW et al. solve this problem in [42] by a continuous transition from the

0 500 1000 1500 2000 2500 1 3 5 7 9 11

Bulk formingBulk forming

Sheet metal formingSheet metal forming

Surface pressure in MPa Ratio of surface enlargement

State of the art

11

COULOMB model in the area of small surface pressures to the shear friction model in the area

of large surface pressures. The frictional shear stress in the Shaw friction law can be

represented by means of a hyperbolic tangent as follows:

𝜏f = 𝑘√𝑡𝑎𝑛ℎ (𝜇 ∙ 𝑝

𝑘)𝑛𝑛

2-12

The parameter n determines the behaviour of the friction law in the transition region. Figure

2-6 shows an overview of all the models mentioned above and compares their estimation

about the progress of frictional shear stress as a function of surface pressure.

Figure 2-6: An overview of COULOMB [37], shear friction [38], OROWAN [41], and SHAW [42] friction

models

However, none of these models considers the surface roughness of the bodies. Actually,

when two nominally flat surfaces are placed in contact, surface roughness causes discrete

contact spots. The total area of all these discrete contact spots constitutes the real contact

area Ar, and in most cases, this will be only a fraction of the apparent contact area Aa. The ratio

of the real contact area to the apparent contact area is known as the real contact area ratio α:

𝛼 =𝐴r𝐴a

2-13

To consider the effect of the real contact area ratio α on the frictional behaviour of bodies,

WANHEIM and BAY proposed in [43] a general friction model:

𝜏f = 𝑓 ∙ 𝛼 ∙ 𝑘 2-14

Where f is the modified friction factor. In this model, the friction shear stress τf is a function

of the real contact area ratio α. Although this is a relatively precise model, it cannot consider

the effects of the lubricant during the process. For this purpose, a complex model for friction

is proposed by BOWDEN and TABOR in [44]. In this model, the frictional shear stress τf is given

by:

𝜏f = 𝛼𝜏a + (1 − 𝛼)𝜏b 2-15

COULOMB Shear OROWAN SHAW

n = 4

n = 2

n = 1

Surface pressure Surface pressure Surface pressure Surface pressure

Sh

ear

str

ess

Sh

ear

str

ess

Sh

ear

str

ess

Sh

ear

str

ess

State of the art

12

Here τa is the average shear stress at contacting asperity peaks, which is related to the real

contact area ratio α, and τb is the average shear stress in the valleys (lubricants pockets), which

can be estimated from the Newtonian viscous behaviour of the lubricant. It is also influenced

by viscosity, pressure, film thickness of the lubricant, and the sliding speed of bodies. If the

friction partners are in a lubricant-free condition, τb will be zero, and the friction shear stress τf

will be only a function of the real contact area ratio. In order to indicate the relevant surface

pressure for different manufacturing processes, KALPAKJIAN and SCHMID showed in [45] the

nonlinear relation between frictional shear stress and surface pressure as a function of the

real and apparent contact areas. Figure 2-7 classified the application range of each process

based on these relations.

Figure 2-7: Friction force as a function of normal force and the range of applications for various

manufacturing processes (the ranges shown are for unlubricated cases) according to [45]

Apart from process parameters, the types of lubricants and their physical properties have

major influences on the lubrication mechanism. In the following, different types of lubricants

used in metal forming are introduced.

2.2.3 Diversity of lubricants in metal forming

There are very many different types of lubricants in use for metal forming operations.

Depending on many parameters, such as the tool and workpiece material, surface roughness

and reactivity, temperature, contact pressure and sliding velocity, these may be classified as

non-emulsifiable oils, oily emulsions, water-soluble synthetics, as well as consistent lubricants

including pastes, greases, soaps, waxes, and solid lubricants [46]. Petroleum-based oil

lubricants incorporate the most common technology. This product category has been used

successfully in metal forming for many decades [47] and is classified into two types: non-

emulsifiable oils and oily emulsions. Non-emulsifiable oils (straight oils) differ in viscosity, and

Surface pressure

Fricti

onal shear

str

ess

Deep drawing

Bending

Stretch forming

Forging, extrusion

Drawing, rolling

Metal cutting

Ar << Aa

Ar < Aa

Ar ~ Aa

State of the art

13

the presence and concentration of lubricity additives and polar and nonpolar additives [48].

There is a vast variety of oil-soluble additives, which may be used to achieve the desired

property of the lubricating oil, such as sulphurs or phosphorous carriers, esters, alcohols, and

sulfonates. The product selection strongly depends on the application of interest. Solvent

diluted oils, including vanishing oils, may be used in aluminium forming or shear cutting.

Medium-viscosity straight oils are used for drawing, extrusion, or fine cutting, while high-

viscosity oils are usually used in heavy-duty forming operations. Emulsifiable oil concentrates

represent an alternative to low viscosity or even solvent diluted oils. They can be diluted with

water to achieve the desired viscosity and performance [49]. At low to medium

concentrations, oil-in-water (o/w) emulsions are formed. Apart from ingredients that are

susceptible to hydrolysis, most additives found in straight oils can also be used in emulsifiable

concentrates. To that extent, there are few delimitations in the choice of additives [50]. While

petroleum lubricants incorporate a very diverse set of products for the metal forming industry,

the category of water-soluble synthetic lubricants may have come to incorporate an even

larger group of products. It has come to represent most water-soluble lubricants that do not

contain mineral oil. This definition includes products composed of vegetable oils, palm

derivatives, and other natural ingredients [51].

When necessary to improve lubricant performance or film stability, thixotropic oils, greases,

waxes, pastes, and even solid lubricants such as graphite and molybdenum disulphide (MoS2)

may be used [52]. Recent studies have shown that dry film lubricants provide better lubrication

in deep drawing when compared with oil-based liquid lubrication [53]. This factor, as well as

the savings for the lubricant used, has helped promote the use of dry film lubricants in the

automotive industry for deep drawing of aluminium and high-strength steel parts [54]. Apart

from the technical and process-based advantages of lubrication, they have many

environmental and economic disadvantages that are described in the next section.

2.3 ECONOMIC AND ECOLOGICAL DISADVANTAGES OF LUBRICATION

From both the economic as well as the ecological points of view, all types of lubricants have

several disadvantages in the sheet metal forming process. For the upcoming process steps

after forming, such as joining, coating, and painting processes, which are required to have a

perfectly clean surface, it is essential to remove the lubricant from the workpiece [55].

Removing the lubricant from the semi-finished parts can be done after some post-treatment

process using many degreasing agents, which are costly and time consuming. Therefore,

using the lubricant in the process leads to an increase in the number of process steps and as

a result, an increase of the cycle time [56]. Focused on the economic point of view, the cost-

saving potential reached by lubricant-free processing can amount to about 2–17 % of the

workpiece-specific production costs, depending on the selected production process [3].

Besides the economic problems, lubricants in the metal forming process have environmental

and health burden disadvantages. Generally, two important manufacturing challenges in the

21st century can be identified as environmental shifts and the deficiency of energy and

State of the art

14

resources. This is not only due to increased production, which is already too high and leads to

an oversupply in many industries, but also is mostly related to how the raw materials and

energy are provided for the production and operation [57]. Based on [58] about 1% of the total

mineral oil consumption is used to produce lubricants. World lubricant demand has remained

flat at around 35 million tonnes per year since 1991, e.g. 37.4 million tonnes of lubricants were

consumed worldwide in 2004: 53% automotive lubricants, 32% industrial lubricants (primary

hydraulic oils) , 10% process oils (metal working fluid), and 5% marine oils [59]. The same

holds true for the EU where the total lubricant demand of about 5.3 million tonnes annually

has stayed unchanged since 1982. Germany is the largest national market for lubricants in

Europe and ranks fifth in global terms with a total lubricant consumption (2.5 million tonnes

yearly), while industrial lubricants account for 1.7 million tonnes and process oils for 0.6 million

tonnes per year, respectively [60]. An estimation from MANG and DRESEL shows that

approximately 50% of the total mineral oil consumption of European countries can be

recovered by recycling and / or further utilisation of waste oil, while the other half is lost to the

environment [59]. MADANHIRE and MBOHWA have characterised in [61] the potential human

health and environmental hazards of widely used classes of lubricating oils because of their

toxic additives. However, lubricants are mostly expected in sheet metal forming for a

successful operation. In the next section, it shall be shown how the lubricants affect the

process stability and the quality of the product through the reduction of tool wear and the

frictional forces between the tool and the workpiece.

2.4 EFFECTS OF LUBRICANTS ON THE QUALITY OF THE DEEP DRAWING

PROCESS

Generally, many parameters have an influence on the deep drawing process and the quality

of the final product. Material properties, tool geometry, blankholder force and friction can be

considered as the main factors affecting the process. Most notably, friction can directly affect

the lifetime of a tool and the efficiency of the process. In the following, some beneficial effects

of lubricants regarding the prevention of wear and galling are pointed out.

One of the most important parameters for the efficiency of a deep drawing operation, i.e. the

production rate, is the operational limits of the forming tools. The lifetime of the tool and the

production speed, have to be carefully balanced to optimise the process efficiency [62].

Increasing the efficiency of the deep drawing operation often goes together with a higher

mechanical load on the forming tools, resulting in higher wear and a shorter lifetime [63].

Wear, defined as the progressive loss of substance from the operating surface of a body,

occurring as the result of relative motion at the surface [64], is one of the most important

sources of process disturbance in sheet metal forming [65]. Tool wear influences the stability

of the equipment, the production speed, and the quality of the products and the product’s

surface [66]. Volume loss of the forming tool and volume loss of the sheet material are two

types of wear in sheet metal forming, resulting from a combination of adhesive wear, abrasive

wear, surface fatigue wear, and tribochemical wear [67]. Among these mechanisms, adhesion

State of the art

15

and abrasion are known as the principal wear mechanisms in sheet metal forming [62].

Adhesive wear in sliding contact is a process where the contact surfaces adhere to each other

so strongly that the atomic bonding forces occurring between the materials on the interface

are stronger than the strength of the surrounding area in both materials [68]. A consequence

of this shear may be the generation of wear particles or a transfer of material from one surface

to the other. Patches of adhered hard material increase the surface roughness of the tool and

when subsequent workpieces are to be formed, the adhered material may indent or scratch

the surface and thereby worsen the surface finish of the product. In sheet metal forming, this

problem is usually referred to galling [69]. The abrasive wear is the removal of material by a

ploughing effect of the hard object into a softer surface [70]. The ploughing can be caused by

a hard asperity on one of the contacting surfaces (two-body wear) or by wear particles in the

contact (three-body wear) [71]. Both the adhesive and abrasive wear behaviour of contact

bodies depend on different influencing parameters. Besides the material properties (chemical

composition, microstructure, surface finish, and hardness), the geometry of the contact area,

surface pressure, and the lubrication system have a major influence on the wear behaviour of

the contact bodies [72]. Increasing the tool wear during the process prevents the control of

strain distribution and as a result the loss of quality and the dimensional accuracy of the

workpiece. Generally, there are three concepts to control the wear in sheet metal forming:

improving the surface roughness of contact bodies to prevent the abrasive wear, changing

the chemistry of either one or both surfaces to lower the adhesion between the sheet and

tool surface, and using the lubricant to separate the sliding surface. Here, the lubricant can

work as a layer of material with lower shear strength than the surfaces themselves [73].

However, the lubricant may not completely prevent the direct contact, although it will reduce

it and may reduce the strength of the junctions formed [72].

2.5 POSSIBILITIES FOR FRICTION REDUCTION IN THE DEEP DRAWING

PROCESS

As the Equation 2-6 implies, there are many parameters, which influence the punch force, like

geometry of the workpiece, sheet thickness, material properties, friction coefficient and

blankholder force. But, the friction coefficient and the blankholder force have a direct influence

on the friction force in the deep drawing process. Reduction of the blankholder as mentioned

in previous sections leads to process instability in the form of wrinkling. As a result of this,

improving the tribological behaviour of the tool through surface treatment can be considered

as the first step towards realisation of a lubricant-free (or minimal quantity lubrication) deep

drawing process. Studies on metal forming without lubricants have shown that results highly

depend on the material combinations to decrease the friction coefficient and wear in critical

parts of the tools [3]. Generally, the different approaches to achieve a lubricant-free or minimal

quantity lubrication forming process can be divided into three categories: ceramic tools [74],

self-lubricating coating systems [75], and hard material coatings [76]. KATAOKA et al.

investigate in [77] the prosperity of ceramic materials like SiC, Si3N4, and Al2O3 (oxide ceramics)

as tool materials for a drawing die within a lubricant-free deep drawing process. However, this

State of the art

16

was by providing a smaller limiting drawing ratio, compared to lubricated processes.

Moreover, a high lubricant-free deep draw ability could be attained with these ceramic

materials for mild steel, whereas no or little improvement could be seen for titanium and

aluminium sheets. TAMAOKI et al. examined in [74] the electro-conductive ceramic tooling like

ZrO2-WC and Al2O3-TiC for lubricant-free deep drawing applications. This type of ceramic

tooling has a higher limiting drawing ratio compared with monolithic ceramic tools in a

lubricant-free condition, but it remains below the lubricated case. Using environmental friendly

self-lubricating materials such as graphite or colloidal suspension can reduce the friction

coefficient significantly; however, they do not attribute further economic benefits to the

lubricant-free deep drawing process [75].

There are many types of surface treatments available to improve the tribological behaviour of

the forming process for lubricant-free applications, which can be categorised in surface coating

and surface functionalisation [78]. The different surface coatings vary in composition,

thickness, and mechanical properties [79]. The two main groups of coating types are deposited

coatings and reactive coatings [80]. Deposited coatings can be obtained by deposition of extra

material on the surface, while the reactive coatings can be obtained, in which a thin layer of

the base surface or the surface of deposited coating is modified through thermal,

thermomechanical or radiation methods [81]. Here, either an alloying element enters to the

base material by diffusion or beam techniques, or the topology of the deposited coating

changes via laser techniques. Some examples are nitriding, carburising, and ion implantation

for direct fabrication of periodic structures on the surfaces.

2.5.1 Surface coating

Since the 1980s, there have been studies on the use of hard material coatings in forming tools

for lubricant-free processes in the context of metal forming [3]. Nowadays, PVD and CVD

coating processes are attractive for industrial applications because the deposition of these

coatings, besides the improvement of the tribological behaviour of the tool, leads to increasing

its lifetime to a large extent [82]. In deep drawing applications, in particular, Titanium Carbide

(TiC), Titanium Nitride (TiN) and TiC/TiN coatings have been used to assure a long lifetime of

the tool because of their high resistance to abrasive and adhesive wear [83]. They provide a

powerful means in those cases where the use of lubricants cannot be completely avoided and

a minimal quantity of lubrication may provide technological and ecological advantages [84].

In recent years, the introduction of the diamond like carbon coatings (DLC) has represented

an attractive solution for the development of lubricant-free forming or processes with minimal

quantity of lubrication due to their excellent tribological behaviour [85]. DLC possesses an

array of valuable properties: an extremely low friction coefficient, abrasion and wear

resistance, exceptional hardness, and high dielectric strength [86]. DLC is considered as an

amorphous material, containing a mixture of sp2 - and sp3 -bonded carbon. Based on the

percentage of sp3 carbon and the hydrogen content, four different types of DLC coatings have

State of the art

17

been identified: tetrahedral carbon (ta-C), hydrogenated amorphous carbon (a-C:H hard and

soft), and hydrogenated tetrahedral carbon (ta-C:H) [87]. JIANG et al. showed in [88] that the

hydrogen content in amorphous carbon films affects their hardness and tribological properties

significantly. Tetrahedral amorphous carbon (ta-C), with a high fraction of sp3-bonded carbons

(up to 80%) and less than 1% hydrogen content, has shown promising tribological and

mechanical properties, such as a low friction coefficient, high hardness (slightly less than that

of diamond) [89], low roughness [90], high inertness, and low mass diffusivity [91]. Figure 2-8

compares the different amorphous carbons, regarding the hydrogen content, their bonding

states and also summarises the properties of ta-C coating. Through evaporation of graphite as

a source of carbon, in the PVD process with arc discharge (vacuum arc PVD) under ultrahigh

vacuum conditions, a pure carbon layer can be reached [92]. Through setting the deposition

parameters like pressure, it is possible to adjust the fraction of sp2 and sp3-bounded carbons

[93]. Because of the vacuum arc process, small graphitic particles (micro-particles), which are

emitted from the carbon source, are incorporated in the coating. This leads to a disturbance

of the layer growth and thus increases the surface roughness [94]. Due to their poor bonding

to the layer composite, these particles reduce the corrosion resistance.

Figure 2-8: Threefold diagrams of amorphous carbon types and the properties of ta-C coating [95]

Pores or pinholes, because of the growth defects are positions with an increasing corrosion

potential. This leads to a local delamination of the coating [96]. Particularly in the case of hard

coatings, such as ta-C, increased roughness decreases the running behaviour of the friction

system. This can result in an increased friction and wear of the counter body [97]. Hence, a

process like brushing is required for roughness reduction of the coated surface.

KUNZE et al. brushed the ta-C coated tools for 5 min using a wire brush (low alloy steel, wire

diameter of 0.2 mm) with a brush force of 50 N. The rotating speed of the wire brush was

6,000 rpm and that of the tool (rotating contrarily) was 150 rpm. They showed in in [98] that

the surface roughness of the coated tools in terms of Ra, Rz, and Rmax can be reduced down to

the roughness of the substrate through a brushing process (see Figure 2-9), while no changes

in coating thickness were observed, as concluded from calotte grinding prior and after

brushing. Furthermore, they showed that an additional brushing does not lead to a further

reduction in surface roughness.

Properties of ta-C:

• Atomic arrangement: amorphous carbon

• Bonding state: up to 80% sp³

• Hydrogen content: less than 1%

• Hardness: 55-65 GPa

• Friction coefficient: 0.12 – 0.19 (without lubrication)

• Ideal for structuring / pattering

sp3

sp2 H

ta-C

Sputtered

a-C:H

ta-C:H

a-C:H

Polymeric

Graphitic C

State of the art

18

Figure 2-9: Average of specific roughness values for the substrate, coated and brushed surfaces [98]

KUNZE et al. examine in [99] the tribological characteristics of the ta-C coated cylindrical

specimens with a layer thickness of 3 - 5 µm via the draw-bend test (see section 2.6) at room

temperature, with a constant drawing velocity of 100 mm/s and a contact surface pressure of

50 MPa. They tested the ta-C coated specimens under lubricant-free conditions and the

uncoated tools under lubricant-free as well as lubricated conditions (lubricating oil “WISURA

ZO 3368”) against workpiece strips from commercially sourced DC04 steel (1.0338) with 1

mm thickness, 20 mm width, and 1000 mm length. Each tool was subjected to 100 draw-

bend tests corresponding to an effective testing distance of 40,000 mm. To assure total

lubricant-free conditions, each strip was cleaned using a citrus-based cleaner, and finally

treated with acetone to remove all traces of pre-lubricants. The resulting average friction

coefficient for the uncoated tool (lubricant-free and lubricated) and the ta-C coated tool

(lubricant-free) are shown in Figure 2-10.

Figure 2-10: Comparison between the friction coefficient of the ta-C coated tool in lubricant-free

conditions and uncoated tools in lubricant-free and lubricated conditions resulting from draw-bend

tests with a range of results

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Ro

ug

hn

ess in

µm

0.11

0.6

Substrate

Coated

4.5

0.11

0.79

3.50

0.84 0.98

4.35

1.07

Brushed

Specific roughness values

Ra Rz Rmax

Substrate material: 1.2379

Coating: ta-C

Coating thickness: 3 μm

Brushing time: 5 min

Brush force: 50 N

Wire rotating speed: 6.000 rpm

Tool rotating speed: 150 rpm

Fricti

on

co

eff

icie

nt

µ

0.17

0.19

0.21

0.23

ta-C coated

(Lubricant-free)

Uncoated

(Lubricant-free)

0.25

Uncoated

(Lubricated)

Drawing speed: 100 mm/s

Contact pressure: 50 MPa

Lubricant: WISURA ZO 3368

Workpiece: DC04 (1.0338)

Strip thickness: 1.0 mm

Drawing distance: 400 mm

Number of repetition: 100

~ 1

2%

0.15

~ 1

9% ~

30

%

State of the art

19

As this diagram indicates, the friction coefficient can be reduced from 0.24 to 0.21 (~12%)

through lubrication. However, coating the tool with ta-C reduces the friction coefficient to

~0.17 (lubricant-free conditions) which is about 30% and 19% less than in the uncoated case

under lubricant-free and lubricated conditions, respectively. These results show that the ta-C

tool coating employed introduces a well-defined boundary layer, which is capable of taking

over the tribological functions of the lubricant. More particularly, because of its disordered

network of carbon atoms [100], changing the fraction of sp2 and sp3-bounded carbons is

feasible with the aim of wear resistance improvement.

2.5.2 Surface micro-structuring

Surface structuring to control frictional forces has been intensively used since ancient times

[101]. For example, during the Tong Dynasty in China, ridged patterns or dimples were placed

on the soles of shoes for labourers to help them work on muddy, slippery ground. For

approximately fifty years, surface micro-structuring has been studied as a means of controlling

friction and wear in sliding contacts [102], because of its relative simplicity for implementation

and as it may be used in conjunction with other lubrication approaches for industrial

applications [103]. FRANZEN et al. showed in [104] the influence of surface micro-structuring

on the friction between the forming tool and the workpiece in comprehensive strip drawing

and deep drawing experiments, for different product shapes considering steel and aluminium

sheets. Generally, from the tribological point of view, the surface should fulfil three main

requirements. The surface has to store and transport lubricant and additionally has to pick up

wear particles. The micro-spaces on the conventionally very smooth tool surface can store

and reserve the lubricant during the process (lubricant pockets), and consequently improve

the tribological properties of the surface. Also in lubricant-free conditions, the wear particles

can be collected between the micro-holes and improve the lifetime of the tools by preventing

the wear and galling [105]. There are different techniques or manufacturing processes of

surface texturing. LASAGNI et al. compared in [106] the different micro-structuring methods

regarding the fabrication speed and structure size (see Figure 2-11). The conventional micro-

milling processes can achieve a relatively high surface structuring speed (~10-2 m²/min), but

with a limited resolution, generally between 15 and 100 µm. Electron and Ion-Beam

lithographic methods have been also used to fabricate high-resolution surface patterns with

feature sizes even smaller than 100 nm, but with a very slow fabrication speed [107]. In

addition, almost all lithographic methods are limited to planar surfaces. A very advantageous

method for micro-structuring is provided by laser processing techniques. Laser-based

fabrication methods offer several advantages due to their remote and thus contactless

operation, their flexibility during materials processing, as well as their precise energy

deposition. Direct Laser Writing (DLW) has been, for example, utilised to structure several

materials with features generally between 1 and 100 μm [108]. However, achieving

resolutions lower than 5 µm is normally associated with an increased technical complexity and

long processing times [106].

State of the art

20

Figure 2-11: Different micro-structuring technologies differing from the surface fabrication speed and

structure size [106]

An innovative solution for high-speed surface patterning of periodic structures in a one-step

process is Direct Laser Interference Patterning (DLIP). Here, periodic structures can be

produced in different materials including metals, ceramics, polymers, and coatings in a single

step process [109]. A significant advantage of this technique compared with other surface

structuring methods is that large areas can be processed within a short period. This method

is particularly suited to fabricate periodic patterns on planar, as well as non-planar surfaces.

Moreover, DLIP enables the fabrication of complex structures by systematically varying the

dimensions of the gratings superimposed upon each other [110]. The DLIP process employs

the interference generated by coherently overlapping two or more laser beams and thereby,

producing a periodic modulation of the laser intensity (see Figure 2-12-A).

Figure 2-12: Direct Laser Interface Pattering; A) schematic of the interference principle, B) line-type

pattern with corresponding two laser beams and C) dot-type pattern with corresponding three laser

beams [111]

0.1 1.0 10 100

10-4

10-3

10-2

10-1

100

101

Multiple-step process

E-Beam lithography

Interference lithography

One-step process

DLIP (objective): 1-5 m2/min

DLIP (2015): 0.1 m2/min

Structure size in µm

Fabrication s

peed in m

2/m

in

Beam 1

(mm-cm)

Beam 2

(mm-cm)

Interference

volume

β

Line-type Pattern with

two laser beams

Dot-type Pattern with

three laser beams

A) B) C)

State of the art

21

The shape of the interference pattern is directly transferred to the material surfaces and

depending on the number of beams used, different geometries are possible [112]. In order to

fabricate one and two-dimensional periodic microstructures on the surface, a two and three

beam interference setup can be utilized (Figure 2-12-B and Figure 2-12-C) [111]. Recent

studies show that the tribological performance of ta-C coatings can be further improved by a

texturing of the surface [113].

In [114], the effect of local transformation of the ta-C coating by means of the DLIP method

from sp3-hybridised carbon into sp2-rich material is investigated. In this study, the ta-C coated

samples of the draw-bend test are structured with the DLIP process using a 532 nm ns-pulsed

laser system. Three different patterns (dot-like, line-like and, cross-like patterns) with a

structure period of Λ = 10 μm are compared with an unstructured ta-C coated tool (reference)

to investigate the impact of local graphitisation of the ta-C film on its wear behaviour, and the

results are shown in Figure 2-13-A. As the diagram indicates, all three types of DLIP structured

samples exhibit much more resistance against wear. The light microscopic images of the

sample shown in Figure 2-13-B reveal that the height of structures after 40000 mm drawing

is not decreased dramatically.

Figure 2-13: Comparison between unstructured and DLIP structured ta-C coating regarding wear

reduction by the draw-bend test; A) relative wear and B) structure height differences before and after

testing [114]

This fact can be explained by the interaction of wear particles in the contact zone and the role

of sp2-bonded carbons on the adhesion strength of the coated layer. For the unstructured

samples, wear particles strongly interact in the contact zone, as they tend to continuously

agglomerate and collapse, resulting in an increase of shear stress in the interface between

the substrate and the coated layer, and as a consequence leads to further delamination of the

Dot-

like p

att

ern

80

60

40

20

Rela

tive w

ear

in %

0

100

Unstr

uctu

red

(refe

rence

)

Lin

e-lik

e p

att

ern

Cro

ss-lik

e p

att

ern

Structure height:

119 11 nm

0 1000

800.0

1.0

0.080

After 40000 mm drawing

Initial state

0 100 µm

Structure height:

94 8 nm

0

0.5

0.7

1.5

µmContact pressure: 50 MPa

Workpiece: DC04 (1.0338)

Drawing distance: 400 mm


Relative wear A) Structures height difference B)

State of the art

22

layer. On the other hand, the patterned surfaces offer the possibility to store wear particles in

the topographic valleys [115]. Therefore, the topology of microstructures can affect the

storage manner of wear particles and then the wear behaviour of the tool. The second reason

for wear reduction through micro-structuring is due to the physical properties of sp2-bonded

carbons in the ta-C layer. Here, the localised sp2-bonded carbons become more intense

towards the substrate-film interface to provide better roughness and adhesion, while the

remaining sp3- bonds have the beneficial effect of friction reduction as shown by MERKLEIN in

[116].

Apart from surface treatment technologies, the reduction of the frictional contact area, as well

as the integral of the contact pressure over the contact area, can also reduce the amount of

frictional forces. This is discussed in Chapter 4 in detail.

2.6 TRIBOLOGICAL TESTING METHODS

Large amounts of test methods for the measurement of the friction coefficient, often referred

to as tribometers, have been developed over the years for a variety of purposes. In this

section, only a small number of tests are presented. These tests are considered either

because of being frequently used, or for representing interesting recent developments in the

field of tribotesting [117].

2.6.1 Tribometer

A tribometer is the general name given to a machine or device used to perform tests and

simulations of friction, wear and lubrication, which are the subject of the study of tribology.

Pin-on-disk and ball-on-disk are two common types of tribometers. According to DIN 50324

[118], in the pin-on-disk (or ball on disk) a pin with a rounded tip is positioned perpendicular to

the other, usually a flat circular disk. A ball, rigidly held, is often used as the pin specimen. The

testing machine causes either the disk specimen or the pin specimen to rotate about the disk

centre. In either case, the sliding path is a circle on the disk surface. The pin specimen is

pressed against the disk at a specified load. In this way, by knowing the pressing force, based

on the COULOMB friction model, the friction coefficient can be measured [119]. However, this

testing method does not consider the surface enlargement during the process in metal

forming. Furthermore, due to the high level of uniform pressure (HERTZIAN pressure) in this

testing method, the friction coefficient cannot be measured under conditions similar to those

encountered in the real forming process. Therefore, in the sheet metal forming process, the

friction coefficient will be evaluated through other testing methods like strip-drawing or the

draw-bend test.

State of the art

23

2.6.2 Strip-drawing and the draw-bend test

For the forming processes with high normal pressure, the dependency of the friction

coefficient on the parameters like pressure, velocity, and sliding direction has to be measured

through a proper friction test method [120]. Strip-drawing and draw-bend friction tests are

regarded as the best candidates for this purpose because of their known capability to change

the contact pressure and sliding velocity [121]. In the strip-drawing test, a strip of metal is

drawn through two friction pads that are pressed together with a defined normal force. The

resultant friction force on the metal strip can be recorded with a metrological device, with

which the COULOMB friction model is determined [7]. However, the non-uniform pressure

distribution at the die edge radius with an arbitrary bending angle can be mapped in the draw-

bend test [122]. The draw-bend test will be performed in a two-step process. First, a strip will

be drawn over a freely turning roller and due to bending and unbending forces, Fb can be

determined as the difference between the pulling and the back tension forces F1 and F2,

respectively. A second strip will then be drawn over a fixed roller, and the corresponding

pulling and back tension forces F1 and F2 can be measured. Figure 2-14 shows the mechanism

of a draw-bend test under deflection angle of θ.

Figure 2-14: Schematic overview of a draw-bend test for the calculation of the friction coefficient

under high non-uniform pressure

Several considerations, each derived based on different sets of assumptions, have been

developed to determine the friction coefficient from the measured forces during a draw-bend

test [123]. FOX et al. developed a model in [124] to express the friction coefficient for a general

deflection angle θ based on an integrated force balance solution (Equation 2-16). In order to

consider the effects of the bending radius rb and the sheet thickness s0, FOX et al. modified

this model in [124] and developed a more detailed expression of the friction coefficient in a

draw-bend test (Equation 2-17). This equation, which was originally obtained from a force

balance, is equivalent with the model of SULONEN et al. [125], which was developed with an

incremental energy balance integrated over a 90° arc (Equation 2-18). A fourth solution

developed by WILSON et al. [126] is based on a macroscopic force balance that assumes the

external forces should be balanced by a roller force due to the effects of the contact pressure

Load cell 1

F2

Tool

Load cell 2

Strip

F1

Contact area

Clamping area

F2 F1

F1

F2

t

rb

θ

State of the art

24

and an average frictional force (Equation 2-19). Their result ignores both the effects of bending

and sheet thickness. It should be noticed that the friction coefficient from Equation 2-19 is

derived from a system force equilibrium and gives the averaged value of the pressure

distribution at the pin / strip contact [127]. Table 2-2 summarises the Equations 2-16 to 2-19.

Table 2-2: Common equations used to calculate the friction coefficient for the draw-bend test

Summary of approaches Equation

Incremental force balance;

ignore sheet thickness;

variable bending angle

𝜇 =1

𝜃ln𝐹1 − 𝐹𝑏𝐹2

2-16


include sheet thickness;


𝜇 =1

𝜃(𝑟b + 0.5𝑠0

𝑟b) ln (

𝐹1 − 𝐹𝑏𝐹2

) 2-17


include sheet thickness;

90° bending angle

𝜇 =1

𝜋(𝑟b + 0.5𝑠0

𝑟b) ln (

𝐹1 − 𝐹𝑏𝐹2

) 2-18

System force balance;

ignore sheet thickness;


𝜇 =2

𝜃(𝐹1 − 𝐹2𝐹1 + 𝐹2

) 2-19

2.7 SUMMARY OF CHAPTER 2

The majority of automotive parts are manufactured by means of the sheet metal forming

technology. The process window in sheet metal forming is highly dependent on the material

flow and restricting parameters. For a given blank shape and final part geometry, it is

necessary to optimise the retention forces applied to the blank in order to control the material

flow and as a result, enlargement of the process window. Friction is one of the most restricting

parameters in sheet metal forming operations. In most cases, it is beneficial to reduce the

friction between the tool and the workpiece to increase the process window, the extension

of the tool life, the prevention of galling and seizure, and the improvement in the surface finish

of the products. There are a number of ways to reduce the friction, like lubricating, improving

the surface roughness, reducing the forces acting on the surfaces, surface texturing and

reducing the contact area.

In the sheet metal forming process, using lubrication reduces the total punch force. However,

from both an economic as well as an ecological point of view, there exists a strong demand

to avoid lubricants within metal forming processes. For the upcoming process steps after

State of the art

25

forming, which are required to have a perfectly clean surface, it is essential to remove the

lubricant from the workpiece. Removing the lubricant from the semi-finished parts can be

done after some post-treatment process, which are costly and time consuming. Therefore,

using the lubricant in the process leads to an increase in the number of process steps. Besides

the increase of process steps, lubricants in the metal forming process have environmental

and health burden disadvantages. Therefore, various green manufacturing strategies under

laboratory conditions have been developed by manufacturers to reduce the friction forces, and

as result reduce the amount of lubrication.

Objectives

26

3. OBJECTIVES

The conservation of required energy, lubricants and other resources for forming processes is

a major engineering challenge from the economic and ecological points of view. Therefore,

developing economically beneficial green technologies - from process and tooling to the entire

enterprise - is one way to insure that future manufacturing systems are sustainable. To

achieve this, innovation in advanced manufacturing is needed.

In sheet metal forming, especially the deep drawing process, lubricants are of importance to

reduce the frictional forces in order to enlarge the process window and increase the lifetime

of tools. So far, the elementary requirements for a nearly lubricant-free deep drawing process

are usually discussed along with tools materials and tool coating to decrease the frictional

force and increase their lifetime. However, a totally lubricant-free deep drawing process has

economic as well as ecological advantages. Figure 3-1 shows the schematic benefits of a

lubricant-free deep drawing process for industrial applications.

Figure 3-1: Advantages of a lubricant-free deep drawing process according to [128]

In order to realise an economically applicable lubricant-free deep drawing process, an adapted

tool design is required to minimise the frictional force during the process. The overall goal of

this thesis is to design a lubricant-free deep drawing process by means of adapted tools, which

ensures the process window despite the absence of lubricants by minimisation of frictional

forces between the sheet and tools, and also the control of material flow during the process.

Since in the absence of lubricants, relevant shear stresses arise because of relative movement

between the sheet and tools, the new tool design must be able to reduce the frictional forces

to the smallest possible extent. In order to decrease the amount of frictional force for a given

friction coefficient, the integral of the contact pressure over the contact area has to be reduced

by means of macro-structured tools. In this way, the contact area between the workpiece and

tools will be reduced to some lines or even some points. As a result of this, the risk of wrinkling

in the unsupported part of the sheet metal increases, because the usually utilised blankholder

force is not applicable. By increasing the geometrical moment of inertia of the sheet, this

effect can be avoided. This can be achieved by immersing the peaks of the blankholder’s

Deep drawing Cleaning Joining Painting

Current state

Deep drawing Joining Painting

• Reduction of process steps

• Reduction of mineral oil requirements

• Saving the cleaning time and disposal costs of lubricants

• Environmental friendly process

Lubrication

Vision of

lubricant-free

production

Lubricant-free

deep drawing

Objectives

27

structure slightly into the valleys of the drawing die and generating a wave structure in the

sheet metal perpendicular to the tangential stress. This method can increase the buckling

stiffness of the sheet metal. Therefore, the macro-structured tool can ensure the process

stability regarding the process limits, i.e. avoiding wrinkling and bottom cracking during the

lubricant-free deep drawing to ensure a process-reliable production.

Generally, the method developed for the lubricant-free deep drawing process should fulfil the

function of usual lubricants by means of the process and tool design. Moreover, an increase

in sustainability should be achieved against conventional process chains.

For understanding the process and its influencing parameters, as well as a time efficient

process design in advance, the process limits should be modelled analytically. Therefore, the

sub-goal of the thesis is to develop an analytical model for the deep drawing of rotationally

symmetric parts based on plasto-mechanical approaches, which can predict the stability of

the process as a function of the tool and workpiece input parameters. In the further

development step, the model should be extended to allow the prediction of process stability

for complex 3D geometries. The feasibility of the process, the results of the analytical models

and the shape accuracy of the deep drawn parts should be verified by means of numerical

simulations as well as experimental tests.

Methodology and approach

28

4. METHODOLOGY AND APPROACH

As already discussed in previous chapters, there exists a strong demand to realise a lubricant-

free deep drawing process without any negative effects on the process window. For this

purpose, the amount of frictional forces has to be reduced. In section 2.1.1, the Equations 2-4

and 2-5 showed that the flange area and the die edge radius of the deep drawing die are the

most critical area regarding the friction. In section 2.5, it was shown that there are several

investigations and examinations which focus on reducing the friction on the die edge radius

of deep drawing tools by means of various surface treatments in order to realise a lubricant-

free process. However, in order to realise a total lubricant-free deep drawing process, the

friction in the flange area should also be minimised. In order to study the importance of friction

on the flange area of a deep drawing process, a number of calculations based on Equation 2-2

to 2-6 are performed. The results show that the share of friction on the flange area Fff from

the total punch force in deep drawing of rotationally symmetric cups may be about 20%,

depending on the dimension of the parts. In this context, the friction coefficient and the

blankholder force play an important role. In order to clarify the importance of these process

parameters on friction reduction in the flange area, Figure 4-1 gives an overview.

Figure 4-1: Reduction of punch force in deep drawing process through 50% reduction of A) friction

coefficient and B) blankholder force

The diagram in Figure 4-1-A shows that for a constant blankholder force (FBH = 100 kN), 50%

reduction of the friction coefficient leads to almost 30% reduction of the total punch force,

while the diagram in Figure 4-1-B indicates that for a constant friction coefficient (µ = 0.15),

50% reduction of the blankholder force can reduce the total punch force by up to 17%.

Therefore, improving the tribological behaviour of the tool through surface treatment as the

current state of the art, and reduction of the total surface pressure, are two methods to

decrease the frictional force. Consequently, a combination of these approaches can lead to

s0 =1.0 mm r0 = 100 mm ri = 50 mm RM = 10 mm σym1 = 300 MPa σym2 = 300 MPa

60,00

80,00

100,00

120,00

140,00

160,00

0 0,05 0,1 0,15 0,2

60,00

80,00

100,00

120,00

140,00

160,00

0 25 50 75 100

Pu

nch

fo

rce

in k

N

Pu

nch

fo

rce

in k

N

60

80

100

120

140

160

Friction coefficient

0.00 0.05 0.10 0.15 0.2060

80

100

120

140

160

Blank holder force in kN

0 25 50 70 100

50 %

30 %

50 %

17 %

FBH = 100 kNA) µ = 0.15B)


29

realising a total lubricant-free deep drawing process. Since reducing the integral of the contact

pressure over the contact area can reduce the amount of frictional force, a novel approach

should be taken towards reduction of the contact area in the flange area. Therefore, the

proposed strategy in this thesis is to minimise the whole contact surface in the forming tool

through macro-structuring. In the following, this is described in detail.

Generally, the frictional force is the product of frictional shear stress and the contact area as

follows:

𝐹f = ∫𝜏𝑓 ∙ d𝐴r 4-1

Here Ff is the frictional force, τf the frictional shear stress (see section 2.2.2) and Ar the real

contact area. This equation can be developed based on the COULOMB friction model:

𝐹f = ∫𝜏𝑓 ∙ d𝐴r = ∫𝜎n ∙ 𝜇 ∙ d𝐴r 4-2

Here, σn is normal stress and μ the friction coefficient. As the Equation 4-2 shows, the normal

stress, friction coefficient and real contact area have a contributing role in friction force during

the process. Based on this equation, a way to reduce the total friction force is reduction of the

contact area. However, this leads to increase of contact pressure locally, because of the

HERTZIAN contact mechanism [129]. But, since the relative high normal contact pressure acts

on only very small zone, the resulting friction force in the integral sum will be minimised.

Based on this consideration, to reduce the contact area between the tools and the workpiece,

macroscopic-structuring of the tools is proposed in this thesis. In this way, the contact area

between the workpiece and the tools will be reduced to a small number of lines, and as a

result the integral of the total contact pressure over the contact area will be reduced compared

with conventional tools. This is depicted schematically in Figure 4-2.

Figure 4-2: Surface pressure in A) conventional and B) macro-structured deep drawing tool

FBH

Conventional Macro-structuredA) B)

FBHFBHFBH


30

In addition, macro-structuring can be used as an analogy to the drawing bead, in order to

control the material flow in the flange area. This aspect is contrary to the reduction of friction,

but is essential for the successful design of a deep drawing process. Thus, the macro-

structuring is provided to reduce the frictional force, as well as for control of the material flow.

For this purpose, two different tooling arrangements can be considered with the capability to

improve the frictional behaviour, as well as to influence the material flow in the flange area of

the deep drawing tool. These differ in the radial offset between the structures of the

blankholder and the drawing die as shown in Figure 4-3. In the next sections, the functions

and application types of each tooling arrangement are discussed regarding the process

stability.

Figure 4-3: Tooling arrangement for macro-structured deep drawing; A) Tip to Tip and B) Tip to Hutch

4.1 MACRO-STRUCTURING: TIP TO TIP

In the deep drawing of parts with relatively complicated geometries and the existence of

tangential compressive stress in the flange area, the risk of wrinkling in the unsupported sheet

metal area will be increased, because the usually utilised blankholder force is not applicable.

To avoid this, the geometrical moment of inertia of the sheet metal should be increased. But,

this is not applicable with the tooling arrangement of Tip to Tip, due to a two-sided contact

between the sheet metal and the tool. Therefore, it can be concluded that this tooling

arrangement can enlarge the process window regarding bottom cracking and cannot prevent

the risk of wrinkling. Nevertheless, this tooling arrangement can be applied only for simple

geometries, where there is no tangential compressive stress in the flange area like the draw

bending of U-Channels.

4.2 MACRO-STRUCTURING: TIP TO HUTCH

As mentioned above, in order to avoid the risk of wrinkling and enlarge the process window

in the deep drawing process with macro-structured tools, the geometrical moment of inertia

of the sheet metal against tangential compressive stresses should be increased. This can be

achieved in the tooling arrangement Tip to Hutch by immersing the blankholder slightly into

the drawing and die inducing an alternating bending mechanism. This creates a wave structure

Tip to Tip Tip to HutchA) B)

Blankholder

Sheet metal

Drawing die


31

in the flange with the desired increased geometrical moment of inertia. Contrary to draw

beads, which are primarily used to control the material flow, macro-structured deep drawing

tools with the Tip to Hutch arrangement can be used to reduce the frictional force due to a

minimal contact area and to increase the resistance against wrinkling. Consequently, based

on fundamental studies regarding the macro-structured deep drawing tool with induced

alternating bending, following positive and stabilising effects can be achieved [130]:

reduction of the contact area up to 80%,

increasing the resistance of the sheet against wrinkling and

possible material flow control.

Figure 4-4 compares schematically the conventional and macro-structured tool regarding the

amount of friction, and also shows how the generated alternating bending can increase the

geometrical moment of inertia of the sheet metal to prevent the wrinkling.

Figure 4-4: Comparison between the A) conventional and B) macro-structured deep drawing process

Since the alternating bending increases the stiffness of sheet metal, it can be expected that

the process window will be enlarged in contrast to the Tip to Tip tooling arrangement regarding

the reduction of bottom cracks as well as wrinkling. Figure 4-5 compares the process window

in the conventional and macro-structured deep drawing process.

Punch

Friction force

σt

Blankholder force

σtCompressive stress:

Punch

Alternating

bending


Friction force

σt

r

z

r

z

σtCompressive stress:

Blankholder forceBlankholder force


32

Figure 4-5: Process window in A) a conventional deep drawing process, B) macro-structuring the tools

with Tip to Tip tooling arrangement, and C) macro-structuring with Tip to Hutch tooling arrangement

However, the geometry of induced alternating bending plays an important role in process

stability. Wavelength and immersion depth are two parameters which determine the

geometry of alternating bending. Wavelength λ is the distance between two supporting points

and immersion depth δ is the height difference between the peak of the die and the

blankholder’s structures as shown in Figure 4-6.

Figure 4-6: Geometrical parameters of induced alternating bending during the process

These two parameters can be used as setting parameters in order to ensure a stable process

for the deep drawing with macro-structured tools. Even by a very low immersion, a plastic

bending mechanism and consequently a further strain hardening in the sheet metal can occur.

That is why besides the geometry part, material properties and the sheet metal thickness, the

radius of alternating bending is an important influencing factor in the stability conditions of

products. Assuming that there are constant mean radius in the sheet metal over the flange

area, the alternating bending radius can be calculated as a function of the wavelength and

immersion depth as follows:

𝑟b =𝜆2

16𝛿+𝛿

4

4-3

Wrinkling

Bottom cracking

Process window

FBH

Drawing ratio

A) Conventional

process

B) Macro-structured:

Tip to Tip

B) Macro-structured:

Tip to Hutch

Wrinkling

Drawing ratio

FBH

Wrinkling

Drawing ratio

FBH

Process window

Process window

Bottom cracking Bottom cracking

Punch Alternating bending

λ

δ

rb: Bending radius, : Bending angle, λ: Wavelength, δ: Immersion depth, rs: Structure radius

δ

rs

rb

rb

θ

λ

rb

r

z


33

Figure 4-7 shows how the bending radius from Equation 4-3 will be reduced by increasing the

immersion depth for different wavelengths. Assuming that the plastic strain begins from

εpl = 0.002, for a sheet metal from steel with a thickness of s0 = 0.6 and 0.8 mm, the critical

bending radius to start a plastic bending mechanism will be 150 and 200 mm, respectively.

This value corresponds to an immersion depth of δ > 0.05 mm. Based on this assessment,

even by small immersion depths and large wavelengths, a plastic bending mechanism will

occur which leads to a further strain hardening in the flange area. However, inducing plastic

alternating bending requires further forming energy, which will be delivered from the punch

force.

Figure 4-7: Change of the alternating bending radius as a function of immersion depth and

wavelength; A) s0 = 0.6 mm and s

0 = 0.8 mm

In other words, in the deep drawing process with macro-structured tools the frictional forces

will be reduced, but in contrast the bending energy will be increased. The energy to induce

the alternating bending supersedes the frictional energy in the process and makes the

lubricant-free process possible. Therefore, in order to reach the same process window of

conventional deep drawing for the macro-structured process, the energy to induce the

alternating bending should be smaller than the energy to overcome the friction. This can be

achieved through a proper tool design. Figure 4-8 compares the energy terms in the

conventional and macro-structured processes.

Figure 4-8: Energy terms in the conventional and macro-structured deep drawing processes

0.0 0.1 0.2 0.3 0.4

Immersion depth δ in mm

0

60

120

180

240

300

Be

nd

ing

rad

ius r

bin

mm

Start of plastic bending

Pure elastic bending

Plastic bending

0.0 0.1 0.2 0.3 0.4

Immersion depth δ in mm

0

60

120

180

240

300

Be

nd

ing

rad

ius r

bin

mm

Start of plastic bending

Plastic bending

Sheet thickness: s0 = 0.6 mmA)

λ = 6 mm λ = 8 mm λ = 10 mm λ = 12 mm

Sheet thickness: s0 = 0.8 mmB)

Pure elastic bending

Ideal forming energy

Bending energy in die radius

Frictional energy in die radius

Frictional energy in flange area

Alternating bending energy in flange area

Conventional Macro-structured


34


In order to realise a lubricant-free deep drawing process, the present frictional forces in critical

frictional areas of the tools, namely the flange area and the die edge radius, should be

minimised to ensure the large process window. There are a number of techniques to reduce

the friction coefficient at the die edge radius by means of surface treatment methods. But,

friction reduction in the flange area of deep drawing tools has not been comprehensively

studied so far. The integrative approach to minimise the frictional forces during the process is

based on macro-structuring of the flange area with different tooling arrangements. The variant

Tip to Tip can minimise the frictional shear stress through the reduction of the contact area.

Nonetheless, this increases the risk of wrinkling which is caused by compressive tangential

stress in the free, non-contact areas of the sheet metal. To avoid wrinkling, the sheet metal

in the flange area should be stabilised by increasing its geometrical moment of inertia. This

can be achieved by immersing the blankholder slightly into the drawing die, inducing an

alternating bending mechanism (variant Tip to Hutch). Therefore, the friction in the deep

drawing process with macro-structured tools will be reduced while the bending energy

increases. Therefore, for a deep drawing process with a large possible process window, an

optimum tool design is required.

Experimental setup

35

5. EXPERIMENTAL SETUP

The realisation of a lubricant-free deep drawing process and verification of the approaches

require a comprehensive experimental test. In this chapter, press machines, tooling and

workpiece material are used, as well as conventional and macro-structured tools are

introduced for the experimental tests.

5.1 PRESS MACHINES

The experimental tests are carried out with two different hydraulic machines, which are

introduced in this section.

5.1.1 Hydraulic press BUP 600

The hydraulic press BUP 600 sheet metal testing machine by Roell Amsler makes it possible

to test sheet metal materials for complex deformation conditions, or to carry out standardised

and approved sheet metal test methods such as the Bulge test, the cupping test, or the

recordings of FLC curves. The machine is equipped with a ViALUX-camera system, which

permits visual evaluation of the local deformation changes by means of visio-plastic tracking.

The machine provides a maximum punching and clamping force of 600 kN, with a maximum

testing speed of 20 mm/s over the maximum ram stroke of 120 mm. The machine can control

the travel of a deep draw piston, its load and the speed of the draw piston continuously with

its hydraulic drive. Figure 5-1 shows the fundamental components of the sheet metal testing

machine.

Figure 5-1: Hydraulic press machine BUP 600

For the deep drawing process, the sheet is clamped between the blankholder and the die. The

punch moves upwards against the clamped sheet metal and draws it into the forming die. The

measurement computer is provided for the evaluation and analysis of the measured forces

and resultant driven forming energy. Regarding the measurement technology, there is a

Testing machine

Piston-cylinder system with blankholder

Optical strain measurement camera

Measurement computer

Tool head (die)

Experimental setup

36

hydraulic pressure sensor integrated in the testing machine, and the pressure will be

converted into force by means of an integrated measuring board.

5.1.2 Hydraulic press Röcher RZP 250

The second hydraulic press which will be used for deep drawing of complex part geometries

is RZP 250 from Röcher Maschinenbau and is shown in Figure 5-2. This press will be used for

deep drawing of complex big size geometries because of its high nominal force (2500 kN).

The table to the right of Figure 5-2 shows the properties of the press. The punch force from

experimental tests is measured indirectly from the hydraulic pressure of the press machine.

The hydraulic cylinder of the press machine has two pistons with mutual hydraulic pressure

and known surface. Measuring the acting hydraulic pressure on each piston, it is possible to

calculate the plunger force. Obviously, there are some disturbance variables like friction

between the cylinder and the pistons, friction on guides (tool and press machine), the weight

of the plunger, tool, etc. In order to eliminate these disturbance variables and obtain the pure

forming force, the measured value should be calibrated based on a comparison with an idle

operation. Through subtracting the measured force of an idle operation from the total force, it

is possible to measure the forming force with a very good accuracy.

Figure 5-2: RÖCHER RZP250 Press

Press plunger

Plunger guide

Measurement computer

Press bolster

Position sensor

Plunger nominal force 2500 kN

Plunger hub 600 mm

Working velocity max. 41 mm/s

Rapid traverse speed 300 mm/s

Die caution 4 x 250 kN

Die caution hub 200 mm

Experimental setup

37

5.2 TOOLING MATERIAL

High-duty, cold work tool steel 1.2379 (EN X153CrMoV12) is widely used in industrial tool

making applications such as deep drawing, punching and blanking of stainless steel, brass,

copper, zinc and hard abrasive materials, generally due to its excellent properties like high

yield stress, high hardness, very good dimensional stability, high compressive strength, good

tempering resistance and high abrasive and adhesive wear resistance. Furthermore, it is a

very good base material for PVD and CVD coating, as well as nitriding due to its secondary

hardening properties [131]. Therefore, it is chosen as the tooling material for experimental

tests in this thesis. The chemical composition of 1.2379 is shown in Table 5-1.

Table 5-1: Chemical composition of the studied tool steel (in wt. %) [132]

Designation to C Cr Mo V

1.2379 1.5 - 1.6 11.0 – 12.0 06 - 0.8 0.9 – 1.0

Tools made by tooling steel 1.2379 can be hardened up to a hardness level of 60-62 HRC

through heat treatment.

5.3 WORKPIECE MATERIALS

Steels and aluminium alloys are widely used materials in today’s industrial applications

involving simple to complex deep drawing processes which require very high formability,

mainly ductility and drawability. In the automobile industry, exterior components such as doors

(frames and panels), front fenders, hoods (frames and panels), roof panels, trunk lids, and rear

side panels are made up of steel and aluminium alloys [133]. Cold-rolled low-carbon steels as

one of the most common forms of steel for forming applications, with the European standard

EN 10130: 2006, are being produced for industrial applications in a broad spectrum of grades,

meeting the specifications as listed in Table 5-2 [134].

The mechanical properties of the individual cold-rolled low-carbon steel grades are

characterised by the yield point and tensile strength, as well as by a guaranteed minimum

fracture strain. Because of the very low carbon content, minimal alloying elements and a

relatively simple ferritic microstructure, they have excellent formability, relatively low strength

and also some of the best weldability of any metal. Therefore, these grade of steels with a

usual thickness ranging from 0.5 to 3.0 mm [135] have long been used for many applications

in the automotive industry, including the body structure, closures, and other ancillary parts.

Among these grades, the fracture strain of DC01 and DC03 is lower than DC04, DC05 and

DC06 (see Table 5-2). Besides that, DC05 and DC06 have lower yield stresses compared to

DC04. Furthermore, the formability of DC06 is highly dependent on the strain rate [136].

Experimental setup

38

Therefore, DC04 was chosen as the workpiece material for the experimental tests in this

thesis because of its excellent formability properties.

Table 5-2: Low-carbon steel grades with mechanical properties and chemical components [134]

Designation to Mechanical properties Chemical composition

EN 10130 EN 10027-2

Material No.

σy0 in

MPa

Rm in

MPa

A80 min

% C % P % S % Mn %

DC01 1.0330 280 270-410 28 0.12 0.045 0.045 0.60

DC03 1.0347 240 270-370 34 0.10 0.035 0.035 0.45

DC04 1.0338 210 270-350 38 0.08 0.030 0.030 0.40

DC05 1.0312 180 270-330 40 0.06 0.025 0.025 0.35

DC06 1.0873 170 270-330 41 0.02 0.020 0.020 0.25

Steel apart, the usage of aluminium in automobiles has been gradually increasing so that

nowadays it is the second-most used material in automobiles and it has the potential to be

used to substitute steel in car body sheet components and become the most-used material

[137]. Among the aluminium alloys, both the 5000 and 6000 series show very excellent

formability and have very good surface quality, though the 6000 series lose their formability

because of their aging effect [138]. Magnesium is one of the most effective and widely used

alloying elements for aluminium, and is the principal element in the 5000 series alloys [139].

When it is used as the major alloying element or combined with manganese, the result is a

moderate to high strength, non-heat-treatable alloy. It is a readily weldable material used in a

wide variety of applications, including pressurised vessels, buildings, transport and the

automotive industry. EN AW-5182 and EN AW-5754 are the principal 5000 series alloys and

are primarily used in the automobile industry [140]. Furthermore, they are also the key alloys

utilised in the production of beverage cans, as well as electrical and computer components

due to its good deep drawing and stretch behaviour [14]. Although both of them have very

similar mechanical properties, EN AW-5754 is usually recommended for relatively high

temperature applications [141]. Therefore, besides DC04, the aluminium alloy EN AW-5182 is

chosen as the second workpiece material for the tests. This material is called AA5182 in this

thesis. The most important mechanical properties and chemical components of AA5182 are

listed in Table 5-3 and Table 5-4, respectively.

Experimental setup

39

Table 5-3: Mechanical properties of AA5182 [142]

Designation to Mechanical properties

Material Nr. Chemical symbol σy0 in MPa Rm in MPa A80 min. in %

EN AW-5182 EN AW-Al Mg4,5Mn0,4 125 275 28

Table 5-4: Chemical components of AA5182 in wt% [143]

Si Fe Cu Mn Mg Cr Zn Ti

0.25 0.35 0.15 0.20-0.50 4.0-5.0 0.10 0.25 0.10

For the numerical simulations and analytical calculations, both testing materials should be

characterised regarding their flow curve by means of uniaxial tensile tests at room

temperature, which corresponds approximately to the same temperature as in the

experimental tests. In order to take the effects of anisotropy and the rolling direction into

account, tensile samples were cut in three directions (0°, 45° and 90° to the rolling direction)

according to DIN 50125 [144]. Each test was repeated five times and the average flow

stressed for DC04 and AA5182 are plotted with a red line in Figure 5-3-A and B, respectively.

Although the tensile test can be used to evaluate the flow curve of the materials, the range of

the equivalent true strain in this test is limited. However, in reality, the materials undergo

higher deformation, so that an equivalent true strain up to 1.0 is required for analysing the

process. In order to extend the flow curves determined from the tensile tests, the curve must

be extrapolated. Therefore, there are various approaches to extrapolate the flow curve [145].

Unlike material models which overestimate the flow stress at a high strain level, VOCE

postulated the existence of a saturation stress at which the rate of hardening should be zero.

Due to its intrinsic saturation character, this model gives a good flow stress description [146]:

𝜎y = 𝜎∞ − (𝜎∞ − 𝜎y0) ∙ 𝑒(−𝑚𝜀pl) 5-1

Where σ∞ is the asymptotic value for saturation strength (ultimate strength), σy0 is the initial

value of isotropic hardening, m and εpl are the characteristic strain constant and equivalent true

strain, respectively. Although this model might underestimate the yield stress at very high

strain levels, since the strain level in the sheet metal forming process is usually around 0.5, it

can better describe the material behaviour than other models [147]. Moreover, in contrast to

many other material models that rely on purely analytical curve fitting, each parameter of this

model has physical significance and can be determined by a simple curve fitting [148]. The

matched VOCE formulation parameters for testing both materials DC04 and AA5182 are listed

in Table 5-5.

Experimental setup

40

Table 5-5: Parameters of the VOCE model for testing both materials

Material σ∞ in MPa σy0 in MPa m

DC04 420 155 7.5

AA5182 390 150 10

Based on fitting these parameters, the flow curves of DC04 and AA5182 are extrapolated

regarding the VOCE material model up to an equivalent true strain of 1.0. Figure 5-3-A and B

shows that there is a good correlation between the experimental data and the extrapolated

curve for both materials.

Figure 5-3: Flow curve of workpiece materials; A) DC04 and B) AA5182

5.4 CONVENTIONAL AND MACRO-STRUCTURED TOOLS

Within the scope of the experimental tests, different deep drawing tools with diverse

objectives were considered. In order to investigate the effect of macro-structuring on friction

reduction as well as the effect of alternating on the springback behaviour of specimens, the

deep drawing tools for draw bending of the U-Channel were used, while to study about the

process stability tools for deep drawing of closed-form geometries like rotationally symmetric

or rectangular cups were chosen.

5.4.1 Tools for draw bending of U-Channels

Draw bending of U-Channels is considered in this thesis to examine the influence of the

contact area on friction reduction in a deep drawing process and the effects of alternating

bending during deep drawing with macro-structured tools on the geometrical quality of the

0

100

200

300

400

500

0 0.2 0.4 0.6 0.8 1

0

100

200

300

400

500

0 0.2 0.4 0.6 0.8 1

Flo

w s

tress in M

Pa

Flo

w s

tress in M

Pa

Equivalent true strain Equivalent true strain

DC04A) AA5182B)

VOCE-model

Tensile test

VOCE-model

Tensile test

Experimental setup

41

parts. Draw bending of U-Channels is chosen for two main reasons: Firstly, there is no ideal

forming as a result of tensile-compressive stress in the flange area, and the total forming

energy consists of bending and friction. Therefore, the effect of the contact area, as well as

alternating bending on the forming energy can be determined directly with this test. Secondly,

this process allows fast responses due to its simplicity, which is suitable for performing a large

number of parameter studies. For this purpose, conventional and macro-structured drawing

tools were constructed from tool steel 1.2379. Figure 5-4 shows the top and side views of

conventional and macro-structured tools with corresponding dimensions schematically. Here,

in the test series, where a calculation of the surface pressure is required (i.e. variant Tip to

Tip), the peaks of macro-structured tools are flattened with a width of 0.5 mm in order to avoid

the HERTZIAN pressure and to be able to gain an analysable condition. Furthermore, the contact

area of both tools were polished to reach a roughness of Ra = 0.3 µm and Rz = 0.5 µm for

comparable testing conditions and to assure the reproducibility of results.

Figure 5-4: Geometry and dimensions of A) conventional and B) macro-structured draw bending of

U-Channel forming tools

5.4.2 Tools for deep drawing of axial symmetric parts

In order to verify the applicability of the macro-structured deep drawing process and its

influence on process stability regarding wrinkling, conventional and macro-structured

rotationally symmetric tools from tooling material 1.2379 were manufactured for experimental

tests. Because of structural reasons, the outer radius of the tools was limited to 113 mm. In

order to reach drawing ratio greater than 2, the inner radius of the tools selected was

ri = 52 mm. Based on preliminary investigations, the die edge radius chosen for both

85.5 mm

65

.5

mm


100 mm 85.5 mm 100 mm

9 mm

0.5 mm

Blankholder

DiePunchPunch

Blankholder

Die

65

.5

mm

Sid

e v

iew

To

p v

iew

x

z

x

y

x

z

x

y

Experimental setup

42

conventional and macro-structured tools was RM = 10 mm. Moreover, a constant wavelength

of λ = 8 mm was considered for the macro-structured tool. Additionally, in order to verify the

transferability of the approach into the non-rotationally symmetric parts, a rectangular tool was

constructed with inner and outer dimensions of 85 mm × 85 mm and 205 mm × 205 mm,

respectively. In order to be able to ensure comparability, a constant wavelength of λ = 8 mm

and a die edge radius of RM = 10 mm were considered for the rectangular tool. In order to

prevent very small straight parts in the rectangular tool, the corner radius should not be very

large. From the other side, a very small corner radius and considering the draw ratio, leads to

a reduction of the outer radius of the sheet metal, and consequently the alternating bending

mechanism cannot take place. Because of these limitations, a corner radius of RC = 20 mm

was chosen for the rectangular tools. Furthermore, the contact areas of all three tools were

polished to get an equivalent surface roughness of Ra = 0.3 µm and Rz = 0.5 µm. Figure 5-5

shows the top view of conventional and macro-structured rotationally symmetric tools, as well

as the macro-structured rectangular tool with corresponding dimensions.

Figure 5-5: Tools for deep drawing of axial symmetric parts; A) Conventional, rotationally symmetric,

B) macro-structured, rotationally symmetric and C) macro-structured, rectangular

Macro-structured

rectangular tool

C)

R20

Tooling material: 1.2379, wavelength: 8 mm, die edge radius: 10 mm, Ra = 0.3 µm, Rz = 0.5 µm

Macro-structured

rotational symmetric tool

B)

R52

Conventional


A)

R52 205 m

m

85 mm

Process modeling and determination of process criteria

43

6. PROCESS MODELING AND DETERMINATION OF

PROCESS CRITERIA

In the deep drawing process with macro-structured tools, the process stability strongly

depends on the selection of process inputs, i.e. wavelength and immersion depth. Therefore,

to design a tool with the greatest possible process window, the process should be analysable

regarding the correlation of influencing factors. This can be achieved through a combination

of analysing methods i.e. trial and error, FEM or developing an analytical model which can

predict the best proper setting parameters. Trial and error methods of testing should be used

only as a final choice, because the process has many drawbacks that make it an unwise choice

in certain situations. Besides that, FEM’s main advantage is that it produces a comprehensive

set of results. However, verification of the results can be sensitive to boundary conditions,

and therefore the model usually needs to be refined repeatedly to give assurance that the

results are reasonably accurate. Subsequently, the results should be verified and optimised

with experimental testing through trial and error methods. Therefore, it is a time-consuming

procedure regarding the tool design in sheet metal forming operations. However, despite

these techniques, an analytical model can be developed for understanding the process

behaviour regarding its response by changing the input parameters in a time-efficient manner.

However, FEM simulation based on the preliminary results of the analytical model might be

required for the final tool design, but it takes much less time. Figure 6-1 compared both

procedures regarding the required time needed for tool design.

Figure 6-1: Comparison of time needed for tool design between two procedures

Because of that, the physical characteristics of the deep drawing process with macro-

structured tools need to be modelled analytically (or semi-analytically). Consequently, based

on the information gained from the model, the criteria for prediction of the process limits

should be developed. The criteria should be able to predict the onset of wrinkling in rotationally

symmetric areas of the deep drawn parts and also to determine the maximum allowable punch

force for prevention of bottom cracks. Furthermore, a criterion is necessary to make a

statement about the geometrical properties and the dimensional accuracy of the specimens

after the forming operation. In the following, the process is modelled and the proposed criteria

are introduced.

FEM Trial and error

Analytical FEM

Time needed for tool design


44

6.1 PROCESS MODELLING: BUCKLING STABILITY

Since in the deep drawing process with a macro-structured tool, the usually utilised

blankholder force is not applicable, understanding the influence of parameters for buckling

stiffness of the sheet metal plays an important role for a stable process. For this purpose, an

analytical model has to be developed to describe the buckling stability of the sheet metal as a

function of geometry and material properties in the most critical areas.

6.1.1 Identification of the most unstable part of the flange area

As discussed in Chapter 2, the tangential compressive stresses cause wrinkling in the

rotationally symmetric parts of the deep drawing process, while the radial tension and the

normal compression stress from the blankholder have a compensating effect. In deep drawing

with macro-structured tools, the induced alternating bending increases the buckling stiffness

of the sheet and prevents the buckling. However, this process provides so far a special case,

in which the peripheral area of the flange (the red area in Figure 6-2) in contrast to the inner

areas (the green area in Figure 6-2) is supported only on one side by the tool and forms a free

end part. Consequently, the free end part of the sheet in deep drawing with the macro-

structured tool is the most critical area regarding wrinkling since (see element 1 in Figure 6-2):

i) the tangential compressive stress (cause of wrinkling) in this area is maximum, ii) the radial

tangential stress (wrinkling compensator) in this area is minimum (compare elements 1 and 2

in Figure 6-2), iii) this part is supported only on one side, while the other parts have two

supporting points, and iv) there is no alternating bending in this area, while the other parts

have a wave structure which increases their buckling stiffness.

Figure 6-2: Stress state on the flange area of the deep drawing process with macro-structured tools.

That is why this part is the most critical part of the sheet metal regarding wrinkling, i.e. if the

process is unstable, wrinkling will initiate from the free end part. Therefore, it is necessary to

control the stability only in this area. In the following, the conditions for stability shall be studied

in the free part of the sheet metal.

1

2 1

2

r

σ

σr

σt

Drawing direction

1

2

r

z

Free end part of sheet

-

+


45

6.1.2 Buckling analysis of the free end part of the sheet

By taking into account that area 1 in Figure 6-2 undergoes no alternating bending, σr and σt are

the present stresses in this region. Considering the force equilibrium condition, the radial

tensile stress σr can be written as [9]:

𝜎r(𝑟) = 𝜎y(𝑟) ∙ ∫d𝑟

𝑟= 𝜎y(𝑟) ∙ ln (

𝑟0𝑟) 6-1

Where σy(r) is the corresponding yield stress at any arbitrary radius r, and r0 is the initial radius

of the sheet. Regarding the TRESCA yield criteria, the tangential compressive stress σt(r) can

be written as:

𝜎t(𝑟) = 𝜎r(𝑟) − 𝜎y(𝑟) 6-2

Having the flow curve of the material, it is possible to calculate the current stress state at the

free end part of the sheet. Modelling the free end of the sheet with a rectangular plate,

supported on one side (simply supported) and free in movement at the other side which

sustain high tangential compressive and low radial tensile stresses, it is possible to perform a

buckling analysis. The first theoretical examination of plate buckling was by BRYAN [149], who

obtained a solution to the problem of a simply supported plate under uniform compression in

1890. Since then, numerous researchers have investigated the local instability in plates under

a wide variety of loading and boundary conditions using many different methods of analysis.

When a compressed plate buckles, it develops out-of-plane ripples, or buckles, along its

length. In the elastic range, the buckled portions of the plate lose their loading, and become

ineffective at resisting further loading, while in the portions of the plate close to the supports,

out-of-plane buckling is diminished, and these parts have post-buckling reserves of strength

and stiffness. The plate as a whole sustains increases in load after buckling, but the axial

stiffness reduces [149]. The post-buckling behaviour of individual thin plates is governed by

two simultaneous non-linear differential equations originally set up by VON KÁRMÁN [150]. For

this aim, a rectangular perfectly flat plate with thickness s0, length and width dimensions a and

b, respectively, was subjected to uniform compressive stress σt and tensile stress σr, which is

supported on one side and free in movement at the other side, and is considered in Figure

6-3:

Figure 6-3: Buckling of rectangular plate under uniaxial compression and tension

ab

s0xy

z


46

The equilibrium equation for the flat plate under uniaxial tension and compression can be

written as:

𝜕4𝑤

𝜕𝑥4+

2𝜕4𝑤

𝜕𝑥2𝜕𝑥2+𝜕4𝑤

𝜕𝑦4=12(1 − 𝑣2)

𝐸𝑠03 (𝜎𝑡

𝜕2𝑤

𝜕𝑥2− 𝜎𝑟

𝜕2𝑤

𝜕𝑦2) 6-3

Where, w denotes the deflection in the z-direction, v and E are POISSON’s ratio and the YOUNG

modulus, respectively. Substituting σr and σt from Equations 6-1 and 6-2 into the Equation 6-3

and taking into account that x and y correspond to the width and perimeter of the free end

part of the sheet in the flange area, this equation make a statement about the buckling

strength of this part. Therefore, this investigation leads to understanding the correlation

between the material properties and structural geometry, and their influence on the stability

of the sheet metal in the flange area. Besides that, for better understanding the influence of

process parameters on the forming energy, this is followed with the analytical method.

6.2 PROCESS MODELLING: FORMING ENERGY

Generally, the load-displacement progress in the deep drawing process is of importance,

because a high level of loading leads to process instability in the form of bottom cracks. The

calculation of the total punch force for the conventional deep drawing process based on the

investigation of SIEBEL was already introduced in section 2.1.1. However, in the deep drawing

process with macro-structured tools, it differs from the conventional process because of

additional alternating bending. In this section, the required forming force is modelled for deep

drawing with a macro-structured tool.

There are several methods to calculate the forming force for the deep drawing process. DOEGE

has compared in [9] the method of SIEBEL [151], BAUER [152] and the “principle of virtual

work”. As Figure 6-4 shows, there is a very good agreement between the experimental results

and the results from the method principle of virtual work.

Figure 6-4: Comparison between the calculated punch force from the method of Siebel, Bauer and

the principle of virtual work with experimental results according to [9]

300

200

100

0

0 20 40 60 80 100

Pu

nch

fo

rce

in

kN

Punch displacement in mm

ExperimentPrinciple of virtual work

SiebelBauer

Material: AA5182

s0 = 1.2 mm

r0 = 210 mm

ri = 100 mm

FBH = 100 kN


47

Since the semi-analytical method based on the principle of virtual work can predict the forces

in the deep drawing process more accurately compared with the SIEBEL formulation (approx.

20% [4]), this method is used in this thesis to calculate the punch force.

The total energy for deep drawing with macro-structured tools Et, consists of the ideal energy

in the flange area Eid, the energy to realise the alternating bending, as well as the bending at

the radius of the drawing die Eb, and the energy to overcome the friction on the contact area

Ef, see Figure 6-5 (compare with Figure 2-2).

Figure 6-5: Individual components of the total forming energy for deep drawing of a rotationally

symmetric cup with a macro-structured tool

The summation of all the energy terms results in the total energy Et:

𝐸t = ∫𝐹t ∙ dℎ = 𝐸id + 𝐸b + 𝐸f 6-4

Ft and dh represent the punch force and punch displacement. Based on the principle of virtual

work, it is possible to calculate the ideal energy for axial symmetric geometries. The principle

of virtual work implies that the external work corresponds to the inner work, i.e.

𝛿𝑊int = 𝛿𝑊ext 6-5

Equation 6-5 indicates that the virtual external work, which results from the external forces, is

equal to the virtual internal work, which is calculated from the existing internal stresses and

the virtual strains. Therefore, in a forming process, the virtual energy for plastic deformation

of a certain volume has to be equal to the product of the according applied force and the virtual

punch displacement. To establish the progress of ideal energy during the process numerically,

it is required to consider the process incrementally. The incremental process evaluation allows

the consideration of the process time and the relevant non-linearities. For this, the whole

process should be divided into small time intervals. To calculate the ideal forming energy Eid,

the frictional and bending energy are not considered, but only the available energy inside the

σt

σr+dσr

r

dα

dr

ri

r0

RM

Ideal forming energy

(Eid)Bending energy

(Eb)Friction energy

(Ef)

ra

s0


48

sheet metal in the flange area. Furthermore, the sheet metal forming process is considered

quasi-static, i.e. the inertial forces are neglected. By an appropriate cut-out volume of the sheet

metal, it is possible to find out the relevant forming forces. Therefore, the principle of virtual

work can be written as follows [153]:

𝛿𝐸id = ∫Ʃ ∙ 𝛿𝑢 d𝐴 = ∫𝜎ij ∙ 𝛿휀𝑖𝑗d𝑉 6-6

Here, Ʃ represent the external stress of the considered cut-out volume, δu the virtual

displacement on the cut-out location, and σij and δεij the internal stress and virtual strain of the

volume, respectively. Figure 6-6 illustrates the considered flange volume, virtual

displacements and also the corresponding radius.

Figure 6-6: Principle of the analytical calculation of ideal forming energy in the flange area according

to the principle of virtual work [153]

Considering the mean vertical anisotropy �̅� according to HILL [154] and by neglecting the

change of sheet thickness during the process, the ideal forming energy in the flange area can

be calculated as [9]:

𝐸id = √�̅� + 1

�̅� +12

∙𝛿𝑉

𝛿ℎ∙ ∫ 𝜎y(𝑟) ∙

1

𝑟d𝑟

𝑟a

𝑟i∗

6-7

Here, δV and δh are the virtual volume change and virtual punch displacement respectively; ri*

is the inner radius of the flange, while ra is the current outer radius of the sheet metal. The

instantaneous yield stress σy(r) of any point in the flange area of the rotationally symmetric

blanks, with an initial radius of r0, can be determined as a function of the current outer flange

radius ra and the corresponding radius of the point r [155]:

𝜎y(𝑟) = 𝑎 [2

√3ln(

√𝑟02 + 𝑟a

2 + 𝑟2

𝑟)]

𝑛

6-8

ri*

δr

r

ra

Ʃδu

Fδh

δu


49

Where a and n are the material constants and can be determined through material

characterisation tests. As Figure 6-5 shows, the sheet metal undergoes an additional

alternating bending in the flange area, besides the double bending over the die edge radius.

The energy for the alternating bending in the structured surface, as well as the bending at the

die edge radius, can also be calculated with the help of the principle of virtual work as follows:

𝐸b = (𝑟M +𝑠02) ∙ √

�̅� + 1

�̅� +12

∙ 𝛿𝑉 ∙ ∫cos(𝜃)

𝑟(𝜃)

𝜃

0

𝜎𝑦(𝜃)d𝜃 6-9

Here s0 is the initial sheet thickness and rb and θ are the bending radius and bending angle,

respectively. The current yield stress σy(r), depending on the radius and the virtual punch

displacement, can be determined through Equation 6-8. Figure 6-7 illustrates the considered

volume for bending over the die edge radius, as well as the present internal and external loads

on the sheet metal in this area. Obviously, the model can be used to compute the required

bending energy for alternating bending in the structured area of the flange through applying

the corresponding bending radius and bending angle.

Figure 6-7: Principle of the analytical calculation of the bending energy, according to the principle of

virtual work [153]

The calculation of frictional energy between the part and the tools is based on the explicit use

of the local pressure and the local frictional force present along the contact length, as originally

pursued by EULER and EYTELWEIN [156]. But, on the die edge radius occur high local stress

peaks, which cannot be considered in that model. Accordingly, in the half-analytical model

based on the principle of virtual work, the frictional energy can also be calculated from the

present stresses [9]:

𝐸f = (𝜇

𝜇 − 1) ∙ √

�̅� + 1

�̅� +12

∙ 𝛿𝑉 ∙ ((1

2𝜃 +

1

𝜇 − 1) ∙ 𝜎y,1 + 𝜃 ∙ 𝜎y,2) 6-10

Where, µ is the friction coefficient, θ is the bending angle and σy,1 and σy,2 are the yield stress

before and after the bending, respectively. With the ability to calculate all the individual energy

RM

θ

δr

δrθ

Ʃδrθ

Fδh

rθ


50

terms for deep drawing with macro-structured tools and superposing them, the correlation

between all the influencing parameters and the forming force can be studied in advance.

6.3 EXTENSION OF THE MODEL FOR DEEP DRAWING OF NON-

ROTATIONALLY SYMMETRIC PARTS

In contrast to rotationally symmetrical parts, there is no uniform stress distribution in the

flange area of irregularly shaped parts along the drawing contour. For example, in the deep

drawing of rectangular cups, the tangential compressive stresses appear significantly in the

rotationally symmetric area, while on the straight parts, act almost exclusively radial tensile

and almost no tangential compressive stresses. In reality, however, the tangential stress in

transition from the rotationally symmetric part to a straight area does not suddenly drop to

zero, but it is reduced continuously in a transition zone. This is show schematically in Figure

6-8-A.

Figure 6-8: Stress state in the flange area of the rectangular cup; A) Distribution of tangential stress

in the flange area according to [157] and B) the stress state in rotationally symmetric and straight

parts of the flange area

During the deep drawing of irregular sheet metal parts, the individual parts of the sheet metal

part undergo different stresses, which can also change during the forming process. Thus,

there should be a distinction between areas where tensile stress states exist and areas where

tensile-compression stress occurs. Therefore, in a first approximation, the flange area can be

subdivided into the rotationally symmetric and straight areas [157], as shown in Figure 6-8-B.

Based on this consideration, the 2D-analytical model developed in this thesis can be extended

for deep drawing of non-rotationally symmetric parts. For this aim, the model can be used to

describe the buckling stiffness of the flange area in the deep drawing of the rectangular cups

with a simplified assumption, in which the rotationally symmetric and straight part of the

flange can be individually analysed. Consequently, the straight zones can be assumed as a

zxy

zxy

Distribution of tangential

stress in flange area

A) Stress state in different

parts of flange area

B)


51

rotationally symmetric part with an infinite radius. However, due to the absence of tangential

stress in the straight area, buckling stiffness is less relevant. Despite buckling stiffness having

less relevance in the straight area, the energy calculation to determine the load-displacement

progress of the process in this area is of importance. In order to make an accurate statement

about the required forming energy for the deep drawing of rectangular cups with macro-

structured tools, it is required to exclude the ideal forming energy from the calculation, i.e.

Equation 6-6.

6.4 DEVELOPMENT OF A CRITERION FOR THE PREDICTION OF

WRINKLING

In order to be able to make a statement about the onset of wrinkling in the deep drawing

process with macro-structured tools, the model developed in section 6.1 can be used. Based

on this model, a criterion to predict the onset of wrinkling is developed within the scope of

this section.

Generally, considering the buckled mode of the plate shown in Figure 6-3, the deflection

geometry of the plate w in the z-direction can be written as:

𝑤 = ∑ ∑ 𝑤𝑚𝑛 sin𝑚𝜋𝑥

𝑎𝑛=1,2,3,…𝑚=1,2,3,…

sin𝑛𝜋𝑦

𝑏 6-11

Here m and n are the number of half sine waves in the buckled mode. It should be taken into

account that, w = 0 at x = 0, y = 0 and y = a. Considering these boundary conditions, the

substitution of Equation 6-11 into Equation 6-3 gives:

(𝑚4𝜋4

𝑎4+ 2

𝑚2𝑛2𝜋2

𝑎2𝑏2+𝑛4𝜋4

𝑏4) =

12(1 − 𝑣2)

𝐸𝑠03 (𝜎t

𝑚2𝜋2

𝑎2− 𝜎r

𝑛2𝜋2

𝑏2) 6-12

Solving the differential equation, the compressive stress σt can be written as:

𝜎t =𝜋2𝐸𝑠0

2 ((𝑚𝑎 )

2+ (

𝑛𝑏)2)2

12(1 − 𝑣2) ((𝑚𝑎 )

2− (

𝜎r𝜎t) (𝑛𝑏)2) 6-13

The lowest value of the plate buckling stress σt,cr in this equation can be obtained when the

plate buckles only in one direction, i.e. in the case of wrinkling n = 1. Therefore, the critical

tangential stress σt,cr can be written as:


52

𝜎t,cr =

𝜋2𝐸𝑠02 ((

𝑚𝑎)2+ (

1𝑏)2

)

2

12(1 − 𝑣2) ((𝑚𝑎 )

2− (

𝜎r𝜎t) (1𝑏)2

)

6-14

This equation gives the critical value of the tangential compressive stress for the onset of

elastic buckling in a plate with given dimensions under uniaxial compression and tension

stress, σt and σr. In relation to wrinkling, m indicates the number of initiated wrinkles. However,

since in the deep drawing process, wrinkling forms a plastic buckling, Equation 6-14 should

be modified for plastic buckling. VON KÁRMÁN [158] has shown that the conventional elastic

bending theory can be extended to cover plastic bending by the substitution for YOUNG’s

modulus E of a plastic buckling modulus, E0:

𝐸0 =4𝐸𝑃

(√𝐸 + √𝑃)2 6-15

Where, P being the slope of the stress–strain curve of the material in simple tension at a given

value of strain; it follows that E0 is a function of strain. In order to apply Equation 6-14 for the

prediction of process stability, the parameter m, which indicates the number of wrinkles in an

unstable process, has to be calculated.

Besides that, In order to transfer this criterion to a deep drawing process with macro-

structured tools, it is required to know the number of wrinkles at the onset of buckling when

the critical stress is reached (m in Equation 6-14). The number of wrinkles in a deep drawing

process depends on the blankholder force. There is no conventional blankholder force by the

free end part of the sheet in the deep drawing process with macro-structured tools. SENIOR

used the energy method in [159] to solve the stability problem in the deep drawing process

for both cases with and without the blankholder. Considering stable equilibria, if the total

energy tending to restore the conditions for equilibrium in a system is greater than the energy

for displacing it, the system remains stable. Equating the two energy values gives the critical

conditions. In the wrinkling of a deep drawn flange due to lateral collapse under the induced

tangential compressive stress, the energy terms involved are of three main types:

EB, the energy due to bending into a half sine wave segment of the wrinkled flange,

Et, the energy due to the tangential compressive forces,

EL, the energy due to the lateral loading of the flange surface.

Due to lateral elastic bending, the energy stored in a half sine wave segment can be written

as:

𝐸B =1

2∫ 𝐸0𝐼

d2𝑦

d𝑥2d𝑥

𝑙

0

6-16


53

Here, E0 is the plastic buckling modulus, I is the moment of inertia and l is the length of a half

sine wave segment and can be determined as follows:

𝑙 =𝑟m𝑚

6-17

Where rm is the mean flange radius and m the total number of waves formed. The energy due

to the tangential compressive forces can also be written as:

𝐸t =𝜎t𝑏𝑠02

∫ (d𝑦

d𝑥)2

d𝑥𝑙

0

6-18

Where b is the flange width. Considering the flange surface as q, the third energy component

EL can be calculated as follows:

𝐸L = ∫ ∫ 𝑏𝑞 d𝑦 d𝑥𝑦

0

𝑙

0

6-19

The critical condition is reached when:

𝐸B + 𝐸L = 𝐸t 6-20

SENIOR solved in [159] the differential equation with respect to the upper and lower limits of

the integral and showed that the number of waves m which the flange would wrinkle is:

1.65𝑟m

𝑟0 − 𝑟i≤ 𝑚 ≤ 2.33

𝑟m𝑟0 − 𝑟i

6-21

Here, rm is the mean flange radius, r0 and ri are the initial radius of the sheet and inner radius

of the drawing die, respectively.

Substituting the upper and lower limits of m from Equation 6-21 into Equation 6-14 leads to

define a safety limit for the tangential stress. The upper and lower limits of the critical

tangential stress σt,cr are plotted in Figure 6-9-A, and this is compared with the tangential stress

at the free end part. As long as the tangential stress during the process lies below the critical

tangential stress σt,cr, no wrinkling is expected and the process will be stable. If the tangential

stress exceeds the critical value and enters into the uncertain zone, the process will be

unstable and wrinkling will be initiated. In order to use this investigation as a criterion for the

prediction of wrinkling in the deep drawing process with macro-structured tools, the progress

of the tangential compressive stress at the free end part of the sheet should be evaluated.

This can be achieved using Equation 6-2. Here, it should be noted that the maximum length

of the free end part is half of the wavelength λ. Because of the strain hardening of the sheet

during the process, the amount of tangential stress increases by each time step. Figure 6-9-B

shows schematically the development of tangential stress, especially at the free end part of

the sheet as a function of sheet radius and time.


54

Figure 6-9: Free end part of the sheet metal; A) Controlling the stability of the process regarding

wrinkling and B) Progress of the tangential stress at the free end part of the sheet during deep

drawing

6.5 DEVELOPMENT OF A CRITERION FOR PREDICTION OF BOTTOM

CRACKS

Generally, fractures occur in the deep drawing process in the bottom corner of the part, since

the material undergoes less plastic deformation and strain hardening, and therefore its

strength is less than other parts. Regarding the theoretical calculation of the maximum

drawing ratio, the forming force which is required to perform the plastification in the forming

area should be compared with the maximum allowable force transmission in the bottom area

of the cup. The greatest possible drawing ratio results from the consideration of equilibrium.

For this purpose, the total forming force based on the developed model in section 6.2 should

be compared with the maximum allowable punch force.

Considering the fracture locus of stainless steel as presented in Figure 6-10, in the space of

the stress triaxiality and the equivalent plastic strain to fracture, it can be observed that the

equivalent plastic fracture strain for plane stress conditions is very close to the uniaxial tension

condition. Therefore, since the stress state in the cylindrical wall of a drawn part is very similar

to the uniaxial stress state, it is possible to use it as a simple failure criterion by comparing it

with the tensile strength of the material.

t1t2t3

tn

Drawing direction

Free end part

B)

Tange

ntial str

ess σ

t

t1

t2

t3

tn

RadiusPunch displacement / Time

Tange

ntial str

ess

No wrinkling

Wrinkling

A)

σt at free end part of

sheet metal

Upper limit of σt,cr

Uncertain zone

Lower limit of σt,cr


55

Figure 6-10: Fracture locus in the space of the stress triaxiality and the equivalent plastic strain to

fracture

Considering Rm as the ultimate tensile strength of the sheet, the maximum transferable punch

or bottom crack force Fbc can be roughly calculated by means of the following equation:

𝐹bc = 𝐴0 ∙ 𝐶R ∙ 𝑅m 6-22

Here, A0 is the cross sectional area of the drawn part and CR is a dimensionless crack factor

for the correction. To identify the crack factor, the model from DOEGE can be used [9]. This

factor can vary from 0.9 to 1.5 as a function of the material and process properties. In order

to determine this factor analytically, it must be assumed that neither the bending, nor the

tangential compressive stress, nor material slippage, nor the friction above the punch edge

radius, influence the transferable force. However, it is to be expected that the punch edge

radius, the sheet thickness, the drawing ratio and above all the current stress state influence

the crack factor. This relationship can be derived using the natural exponential function as

follows [160]:

𝐶R = 𝑒𝜑𝑡 6-23

Here φt is the tangential compression factor at the outlet of the punch edge radius and can be

determined as follows [160]:

𝜑𝑡 = ln

√𝑟p𝐴𝑠0(𝑟i𝑟p− 1) (𝐴2 − 1) + (1 − cos𝛼)

23𝑟p𝑠0(𝑎3 − 1) + (

𝑟i𝑟p− 1)

2

𝑟i𝑟p− 1 + sin𝛼

6-24

with

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.2

0.4

0.6

0.8

1.0

1.2

ɳ = σm / σeq

Eq

uiv

ale

nt

fractu

re s

train

Uniaxial tension

ɳ = 0.33

Plane strain

ɳ = 0.5

Biaxial tension

ɳ = 0.66

σI

σII

VON MIESES

TRESCA

Biaxial tension

Plane strain

Uniaxial tension

YIELD surface


56

𝐴 = (𝑠0𝑟p+ 1) 6-25

Where, rp and α are the punch edge radius and the deflection angle of the punch edge radius,

respectively. Comparing the bottom crack force Fbc as a criterion for the fracture with the

calculated punch force, which is computed through the energy method, it is possible to make

a precise and explicit statement about the stability of the process regarding bottom cracks.

Obviously for a stable process, the punch force must not exceed the maximum transformable

force or the bottom crack force Fbc.

6.6 DEVELOPEMENT OF A CRITERION FOR PREDICTION OF

DIMENSIONAL ACCURACY OF THE PARTS

The deep drawing of sheet metal induces complex deformations, which in different regions

of the sheet metal can have vastly different accumulated plastic strains [161]. Upon unloading

by removing the tools, springback occurs which can change the shape accuracy of the part

[162]. Springback is caused by non-uniform residual stresses, through the sheet thickness of

the formed part, that create a bending moment, which causes a distortion of the workpiece

upon unloading. Springback strains are almost completely elastic; non-linear recovery strains

are usually small but, in extreme cases, they can amount up to 10% of the total recovery [163].

Therefore, reliable prediction of springback is of importance during the tool design. Nowadays,

engineering guidelines and finite element software are used in the design process to predict

the amount of springback in sheet metal forming. Based on this prediction, the tools’

geometry and process parameters are modified to obtain the desired product shape [164].

However, the current accuracy of springback prediction is not sufficient [165].

The fact that materials are subjected predominantly to tangential compression as long as they

are in the flange area, followed by radial tension for the inward cup side and radial compression

for the outward cup side during the bending over the die edge radius, results in complex strain

path changes in the sheet. Additionally, in the deep drawing process with macro-structured

tools, further strain hardening as a result of alternating bending decreases the amount of

elastic strains within the sheet thickness. Moreover, for some materials, the load change

during the alternating bending leads to a kinematic hardening, see Figure 6-11. Due to the

closed-form shapes and accordingly high structural stiffness, the deep drawn parts contain

high residual stresses, which are strongly dependent on the material hardening behaviour.

Since most of them will be cut out for the follow-on operations, the residual stresses lose

their balance, which leads to a dimensional inaccuracy in the form of springback in the part.

The positive effect of the increased retention force for the compensation of springback

through increasing the blankholder force in the conventional deep drawing process is known

from the literature [166]. However, since in the lubricant-free deep drawing process the usually

utilised blankholder force is not applicable, it can be performed geometrically through retention


57

force results from alternating bending. Based on this consideration, developing a criterion to

predict the amount of springback due to the release of the residual stresses after cutting out

the specimen in the deep drawing process with a macro-structured tool is essential. For this

purpose, a half-analytical criterion is developed in this section.

Figure 6-11: Influence of alternating bending on the kinematic hardening behaviour of the workpiece

A way to quantify the circumferential residual stress in deep drawing cups is called ring

splitting (Figure 6-12), a method introduced by SIEBEL and MÜHLHÄUSER in [167]. Ring splitting

is one of the most common springback-measurement tests for the deep drawing of

symmetrical parts, which offers an opportunity to evaluate the amount of residual stress in

the specimen. In this test, a ring (see Figure 6-12-A) will be cut out from the wall part of a

deep drawn cup and subsequently split open. The tangential residual stress can then be

determined from the ring opening chord upon splitting.

Figure 6-12: The ring splitting method to determine circumferential residual stress in deep drawing

cups; A) Cutting area in a rotationally symmetric deep drawing cup for the splitting test, B) springback

in the split ring and C) tangential residual stress at the inner and outer cup surfaces

Radius

Tan

ge

ntial

str

ess

Radius

Drawing direction

Top side of sheet metal

Bottom side of the sheet metal

Middle of the sheet metal

Isotropic hardening

Kinematic hardening

Isotropic expansion

of yield surface

Kinematic expansion of

yield surface σII

σI

σ

ε

σIso

σy0

- σKin

- σIso

Rad

ial

str

ess +

0

-+

0

-

Rotational symmetric

cup for ring splitting test

Change in opened ring

curvature

Δ

r2

r1

Tangential stresses at

outer and inner surface

A) B) C)


58

To determine the tangential residual stress in the wall of the cup, the ring opening gap Δ can

be approximated with a circular arc, as shown in Figure 6-12-B:

2𝜋𝑟2 = 2𝜋𝑟1 + ∆ 6-26

or:

𝑟2 − 𝑟1 = ∆/2𝜋 6-27

Here r1 and r2 are the mean radius of the intact and split ring. The maximum tangential elastic

bending strain due to the springback at the outer and inner surfaces of the intact ring 휀max,outeb

can be determined from the bending theory as follows:

휀max,outeb =

𝑠02𝑟1

−𝑠02𝑟2

=𝑠02(𝑟2 − 𝑟1𝑟1𝑟2

) 6-28

While the maximum tangential elastic bending strain at the inner surface 휀max,inneb is:

휀max,outeb =

𝑠02𝑟2

−𝑠02𝑟1

=𝑠02(𝑟1 − 𝑟2𝑟1𝑟2

) 6-29

By substituting the Equations 6-28 and 6-29 into 6-26, the elastic bending strain can be written

as:

휀max,outeb =

𝑠02(∆

𝑟1𝑟2) 6-30

and

휀max,inneb = −

𝑠02(∆

𝑟1𝑟2) 6-31

Since Δ/2π ≪ r1, it can be assumed that r1 ≈ r2, therefore:

휀max,outeb =

𝑠04𝜋(∆

𝑟12) 6-32

and

휀max,inneb = −

𝑠04𝜋(∆

𝑟12) 6-33

Considering the Equations 6-32 and 6-33, the maximum tangential elastic bending stress at

the outer and inner surfaces can be calculated as follows:


59

𝜎max,outel =

𝐸 ∙ 𝑠04𝜋

(∆

𝑟12) 6-34

𝜎max,innel = −

𝐸 ∙ 𝑠04𝜋

(∆

𝑟12) 6-35

As the Equations 6-34 and 6-35 show, the residual stresses are tensile at the cup outer surface

and compressive at the inner surface. Figure 6-12-C shows these stresses schematically. As

long as there is no springback in the workpiece, because of equilibrium conditions, the bending

moment should be zero due to tangential stresses over the sheet thickness in the wall part of

the cup (intact ring) [168]:

𝑀b = 𝑀bres −𝑀b

eb = 0 6-36

Here, 𝑀bres is the bending moment because of the residual stress in the intact ring and 𝑀b

eb is

the induced elastic bending moment which is necessary to close the split ring. Since 𝑀beb is

an elastic bending moment, considering the equilibrium conditions, it is symmetric about the

natural axis, i.e. over the sheet thickness and can be written as a function of its maximum

value 𝜎maxeb :

𝑀beb = 𝑏 ∙ ∫ 𝜎max

eb ∙2𝑥

𝑠0

𝑠02

−𝑠02

∙ 𝑥 ∙ 𝑑𝑥 6-37

𝑀beb =

4

3𝑏 ∙ 𝜎max

eb ∙ (𝑠02)3

6-38

Substituting Equation 6-38 in 6-36 and using the Equation 6-34 or 6-35, the opening gap Δ due

to springback can be derived as follows [168]:

∆=24 ∙ 𝜋 ∙ 𝑟1

2 ∙ 𝑀bres

𝐸 ∙ 𝑠04 6-39

To find the opening gap for a given deep drawn part, the residual bending moment in the intact

ring 𝑀bres has to be calculated by means of FEM-simulations. This way it is possible to evaluate

the amount of residual stress in a part deep drawn by a macro-structured tool and compare it

with the reference samples.


For better understanding the physical behaviour of the deep drawing process with macro-

structured tools, and also to find out the cause-and-effect relationship between the process

parameters, an analytical model was developed within the scope of this chapter. The model

describes the buckling stability of sheet metal in the flange area and also can determine the


60

load-displacement progress of the process as a function of the process input parameters. In

this chapter, it was shown that the 2D model can be extended so that the deep drawing of

non-rotationally symmetric parts can also be analysed. For this purpose, the flange area should

be subdivided into rotationally symmetric and straight areas, and they should be subjected to

analysis individually. Additionally, for a time efficient prediction of the process limits, some

criteria were devised based on the developed model. With the help of the developed criteria,

the process limits, i.e. bottom cracking and wrinkling, can be predicted based on the energy

method and buckling analysis, respectively. Moreover, a criterion was developed to evaluate

the amount of springback in the wall part of the deep drawn specimens.

Results

61

7. RESULTS

In order to control the applicability of the proposed approach and verify the developed model

and criteria, several experimental as well as numerical tests were carried out and their results

are shown within the scope of this chapter. Experimental tests were performed with the

experimental setups as introduced in Chapter 5, while all numerical simulations were carried

out with the commercial FEM-Software simufact.forming. The results show an overview

about the effects of macro-structuring on process characteristics, e.g. friction, stability and

springback behaviour. Consequently, the developed process model will be verified, as well as

the analytical criteria to determine the process limits and predict the dimensional accuracy of

the part.

7.1 INVESTIGATION ABOUT THE EFFECTS OF MACRO-STRUCTURING

ON PROCESS CHARACTERISTICS

As mentioned above, in this test series the influence of the contact area on friction reduction

is examined. Furthermore, the influence of alternating bending on the process stability and

compensation of springback is investigated. For this end, numerical as well as experimental

tests are performed.

7.1.1 Investigation about friction reduction through macro-

structuring

In Chapter 4, it was discussed how the macro-structuring of tools can reduce the integral of

the surface pressure over the contact area. In order to control the feasibility of this approach

for real applications, the draw bending of the U-channel as a forming process is considered in

this section. Here the conventional and macro-structured tools are used, which were

introduced in section 5.4.1. The tests are carried out with the hydraulic press machine BUP

600 at room temperature, with a constant drawing velocity of 10 mm/s, which corresponds to

the deep drawing process. The conventional and macro-structured tools were subjected to

lubricant-free, as well as lubricated conditions. For the lubrication, the industrial mineral oil-

based lubricant “WISURA ZO 3368” was used. This lubricant, which has some additional

additives like phosphor, is commonly used in the field of metal forming, especially for the deep

drawing of aluminium and stainless steel alloys [169]. Workpiece strips were cut from DC04

steel of 1 mm thickness and 60 mm width to have a good manageability and 280 mm length

to realise a relatively big drawing depth with the existence of the remaining flange area. Each

strip was cleaned using a citrus-based cleaner and finally treated with acetone to remove all

traces of pre-lubricants. All tests were repeated five times in order to get consistent

information about the statistical repeatability of the results. In order to investigate the effect

of the contact area on friction properties, an experimental matrix was defined. Within this

Results

62

matrix, two constant initial surface pressures PTotal = 5.1 and 6.8 MPa were considered for the

draw-bending of the U-Channels, which correspond to the deep drawing process. The results

of the tests are presented in Figure 7-1. As the results show, 28.1 kN is required for the draw-

bending of a U-Channel with a conventional tool in lubricant-free conditions under the initial

surface pressure of PTotal = 5.1 MPa. By increasing the surface pressure to 6.8 MPa, the

increased frictional force prevents the material flow on the flange area, which finally leads to

necking and consequently the tearing of specimens. Lubricating the specimen with WISURA

ZO 3368 improves the frictional behaviour of the process and decreases the punch force to

23.5 kN under PTotal = 5.1 MPa (approx. 16% reduction compared with lubricant-free

conditions). However, the results show that through lubricating, the specimen will not tear

under the initial surface pressure of 6.8 MPa. Although reducing the contact area by means of

macro-structured tools (from 5850 mm2 for a conventional tool to 327 mm2 for a macro-

structured tool, i.e. approx. 94% reduction) leads to an increase of local surface pressure in

the contact zones, based on Equation 4-2, this results in a drastic reduction of frictional forces.

This statement can also be verified with the experimental results. As shown in Figure 7-1, the

average punch force for the draw-bending of U-Channels in lubricant-free conditions with

macro-structured tools under the initial surface pressure PTotal = 5.1 and 6.8 MPa is 6.1 and

6.6 kN, respectively (approx. 74% and 77% reduction compared to the conventional process

in lubricated conditions). Furthermore, using the macro-structured tools in lubricated

conditions reduce the punch force even further, i.e. to 4.9 and 5.1 kN under an initial surface

pressure of PTotal = 5.1 and 6.8 MPa, respectively. As the diagram shows, the deviation of the

results is almost proportional to the average values. This reveals that there are no systematic

errors in the test and measurement procedures.

Figure 7-1: The required punch force for forming the U-Channel with conventional and macro-

structured tools in lubricant-free as well as lubricated conditions

Pu

nch

fo

rce

in

kN

0

5

10

15

20

25

Surface pressure PTotal = 5.1 MPa

Surface pressure PTotal = 6.8 MPa

Sheet material: DC04

Sheet dimension: 285 60 1 mm

Lubricant: WISURA ZO 3368

Punch speed: 5 mm/s

Die edge radius: 10 mm

Ra = 0.3 µm, Rz = 0.5 µm

Number of repetitions: 5

Lubricant-

free

Lubricated

30

35

Te

ari

ng

28.1

6.1 6.6

23.5

28.9

Conventional Macro-structured

9 mm 0.5 mm

Lubricated

4.9 5.1

Lubricant-

free

Drawing Drawing

Results

63

The results reveal that macro-structured tools can be used to reduce the contact area between

tools and the workpiece, and this leads to a decrease in the amount of frictional shear

stresses, and as a result the total frictional force in the process. In the next section, the effects

of alternating bending on the process stability for the deep drawing of rotationally symmetric

and rectangular cups are studied.

7.1.2 Investigation about process stabilisation through macro-

structuring

In Chapter 4, with the help of Equation 4-2, it was shown that macro-structuring tools reduce

the frictional shear stress, and as a result the total punch force in the sheet metal forming

process. It was also discussed that the amount of the immersion depth and the wavelength

as two important process input parameters, which define the geometry of alternating bending,

have a great influence on the process stability regarding wrinkling and bottom cracking. In

order to investigate the sensitivity of the process window regarding the process input, various

numerical simulations were performed. For this purpose, the blanks from DC04 were

subjected to numerical simulations. For this end, sheet metal with a 90 mm radius and

1.0 mm thickness was chosen for 3D simulation of a rotationally symmetric deep drawing

process. In order to reduce the calculation time, a quarter of the workpiece with two

symmetric planes was subjected to the FE-simulation. The inner radius of the tool is 50 mm

so that a drawing ratio of β = 1.8 can be realised. Here, solid shell elements (element size:

1 mm with 5 integration layers per thickness and one integration point per layer) were used.

The variant matrix studied comprises 4 different wavelengths and five different immersion

depths. The parameter limits of the simulation matrix were chosen regarding the process

window, so that the intended wrinkling and bottom cracking occurred. For comparison, a

series of simulations were also carried out with a sheet thickness of 0.6 mm. The calculations

were performed with and without friction in order to be able to analyse the frictional

component and the bending component separately from each other. The numerical results

shown in Figure 7-2 confirm basically the hypotheses of the analytical considerations regarding

the influences of wavelength and immersion depth.

Red marked areas indicate the occurrence of bottom cracks, orange areas indicate the

appearance of wrinkles. Due to the lower buckling stiffness, the 0.6 mm sheet has a higher

sensitivity to wrinkling than the 1.0 mm sheet, which reduces the process window (green

area). Furthermore, as expected, the simulations show that friction increases the risk of

bottom cracks and therefore reduces the process window.

Results

64

Figure 7-2: Results of the numerical parameter analysis of the process window

Moreover, in order to verify this state and control the applicability of the macro-structured

deep drawing process, similar test series were performed experimentally. For this purpose,

the tools introduced in section 5.4.2 were used. Sheet metals from the workpiece material

DC04, which was introduced in section 5.3 with two different thicknesses of s0 = 0.6 and

1.0 mm, were cut for the deep drawing process. Figure 7-3 shows the geometry and

dimensions of the tools and the workpieces.

Figure 7-3: Geometry of the sheet metals used for the control of process stability

Within the scope of the experimental matrix, the stability of the rotationally symmetric deep

drawing process with macro-structured tools was compared to the conventional process. Both

macro-structured and conventional processes were carried out with the use of a spacer ring.

Wavelength in mmImm

ers

ion d

epth

in m

m

6 8 10 12

0.0

0.1

0.2

0.3

0.4

Wavelength in mmImm

ers

ion d

epth

in m

m6 8 10 12

0.0

0.1

0.2

0.3

0.4

Wavelength in mmImm

ers

ion d

epth

in m

m

6 8 10 12

0.0

0.1

0.2

0.3

0.4

Wavelength in mmImm

ers

ion d

epth

in m

m

6 8 10 12

0.0

0.1

0.2

0.3

0.4

s0 = 1.0 mm (µ = 0) s0 = 1.0 mm (µ = 0.1) s0 = 0.6 mm (µ = 0) s0 =0.6 mm (µ = 0.1)

Faultless Wrinkling Bottom cracking

Ø103

Ø225

Macro-structured


A)

Ø200

85 mm

205 mm

Macro-structured

rectangular tool

C)

R20

35

mm

160 mmØ103

Ø225

Ø200

Conventional


B)

Blank Tool Blank Tool Blank Tool

Tooling material: 1.2379, Wavelength: 8 mm, Die edge radius: 10 mm, Ra = 0.3 µm, Rz = 0.5 µm

Results

65

The ring can be used to adjust the amount of the immersion depth, i.e. δ = 0.2 and 0.4 mm in

the macro-structured process. In the conventional process, the height of the ring was set to

be equal to the sheet thickness. Pressing the blankholder with a full load to the spacer ring

keeps the space between the blankholder and the drawing die constant during the process.

Consequently, in order to control the feasibility of the new developed process for more

complex geometries, the deep drawing of rectangular cups was performed with immersion

depths of δ = 0.2 and 0.4 mm. As Figure 7-4-A and Figure 7-4-B show, no wrinkling can be

seen in the workpiece made by the macro-structured process, which reveals that alternating

bending can stabilise the sheet metal against wrinkling.

Figure 7-4: Results of the experimental tests for the control of process stability regarding wrinkling

by deep drawing with macro-structured tools

Unlike the macro-structured process, there was some wrinkling in the workpiece made by the

conventional process, as shown in Figure 7-4-C. Obviously, thinner materials are more

unstable against wrinkling in the deep drawing process. Therefore, in the next step, sheet

metals with a thickness of s0 = 0.6 mm were chosen for the tests to control the applicability

of the process for thin sheets. The results show that there was almost no wrinkling in the

workpiece made by the deep drawing process with macro-structured tools and an immersion

depth of δ = 0.2 mm, as shown in Figure 7-4-D. However, increasing the immersion depth to

ConventionalMacro-structured

δ = 0.2 mm δ = 0.4 mm

Sh

eet

thic

kness

s 0=

1.0

mm

Sh

eet

thic

kness

s 0=

0.6

mm

Sh

eet

thic

kness

s 0=

1.0

mm Lubricant-free conditions

Workpiece material: DC04

Wavelength: 8 mm

Drawing clearance: 0.5 mm

Drawing speed: 10 mm/s

A) B) C)

D) E) F)

G) H)

Results

66

δ = 0.4 mm leads to bottom cracks, as shown in Figure 7-4-E. It is because of the increased

alternating bending energy in the flange area and consequently the increasing total punch

force, which should be transferred to the bottom part of the workpiece. The stress caused by

the punch force over the cross sectional area of the thin part exceeds the ultimate tensile

strength of the workpiece, and this leads to the fracture of the material. Testing the same

material with the conventional process, results in a number of wrinkles in the part, as shown

in Figure 7-4-F. As mentioned above, to test the more complicated geometries with the macro-

structured deep drawing tool, rectangular tools are considered. Here, the stress state in the

transition area, where the tangential stress decreases in the corners toward the straight parts

is more complicated and the risk of wrinkling is higher. For that end, the sheets with a

thickness of s0 = 1.0 mm were subjected to deep drawing with macro-structured tools. The

results, which are depicted in Figure 7-4-G and Figure 7-4-H, show that there is no significant

wrinkling in the workpiece and that the stability of the process is guaranteed. Summarising

the results of the experimental tests, it can be concluded that the induced alternating bending,

during deep drawing with macro-structured tools, increases the process stability through

increasing the geometrical moment of inertia. Therefore, in this way, it is possible to enlarge

the process window even more than with the conventional process. However, it should be

noticed that excessive immersion depths can lead to bottom cracks in the workpiece.

Therefore, process stability strongly depends on the process input parameters. Hence, to have

the greatest possible process window, the optimum values of wavelengths and immersion

depths should be chosen based on the criteria developed in Chapter 6.

7.1.3 Investigation about springback reduction through macro-

structuring

As already discussed in section 6.6, the alternating bending mechanism in the deep drawing

process with macro-structured tools can lead to a kinematic hardening effect of some

materials. Because of this effect, there is a need to investigate the effects of process

parameters on the springback behaviour of the workpiece in the newly developed deep

drawing process. In this section, the effect of alternating bending is studied, as well as the

resulting kinematic hardening on the springback behaviour of the deep drawn parts in the axial

and tangential directions. To get information about the effects of materials on springback in

deep drawing with macro-structured tools, two different types of industry-relevant materials,

the alloy steel DC04 and the aluminium alloy AA5182, were subjected to numerical and

experimental tests. For this purpose, draw bending of a U-Channel was considered. This

method is attractive for these test series, because the level of springback due to the 90°

bending in the axial direction is relatively large and it can easily be measured. The sensitivity

of springback to basic parameters, such as tool radius, sheet thickness, geometric parameters

of the tools, mechanical properties of the sheet materials and friction parameters, is usually

studied by means of this technique [170]. Furthermore, the U-Channel parts are common

elements in industrial sheet metal forming. They appear in many auto-body cover panels like

Results

67

side members and beams. To achieve that, the process without alternating bending

(δ = 0.0 mm) is compared to the process with alternating bending and an immersion depth of

δ = 0.2 mm in draw bending of the U-Channel. The immersion depth of 0.2 mm is chosen

because it corresponds to the stable deep drawing process. Furthermore, in order to examine

the effects of kinematic hardening on springback of the workpiece during the process, the

materials are considered with pure isotropic, pure kinematic and also a combined hardening

behaviour in the numerical simulation. The BAUSCHINGER coefficients of each material for the

combined hardening rule with a plastic strain of 0.3 are taken from literature [171] and listed

in Table 7-1.

Table 7-1: BAUSCHINGER coefficients of testing materials [171]

Material BAUSCHINGER coefficient

DC04 0.60

AA5182 0.85

Here, in the case of 100% isotropic hardening, the BAUSCHINGER coefficient is equal to 1 and

by 100% kinematic hardening it is equal to zero. The process was simulated with a 2D FEM-

Model with plane strain solid elements (7 elements across the sheet thickness of 1.0 mm)

and a constant friction coefficient of µ = 0.15 using the COULOMB friction model. The friction

coefficient was chosen based on the investigations introduced in Chapter 2. For the

experimental tests, the setup applied in section 7.1.1 with the Tip to Hutch arrangement was

used. All the tests were repeated five times in order to get sufficient information about the

statistical repeatability of the results. The springback behaviour of U-Channels can be

characterised regarding the final part geometry with three parameters: the angle between the

bottom and the wall β1, the angle between the wall and the flange β2, and the radius of

curvature of the sidewall ρ. As β1 and β2 increase and ρ decreases (the curvature of sidewall

increases), springback increases. Figure 7-5 shows the springback geometry of a U-Channel

cross section.

Figure 7-5: Schematic Overview for springback geometry for a U-Channel [172]

β2

ρ

x

y β1

Results

68

Since β1 and β2 are usually very close together and there is almost a linear relation between

the sidewall curvature and deflection angles, β1 was chosen as a scale to study the effects of

the process parameters on springback in this section.

Experimental results reveal that the springback of workpieces will be reduced through

generating an alternating bending mechanism, because of the induced tensile force in the

flange area. The geometries of the formed parts which are depicted in Figure 7-6 show that

regardless of the material, the springback in the process with alternating bending (δ = 0.2 mm)

is always less than the process without alternating bending (δ = 0.0 mm).

Figure 7-6: Springback behaviour of the workpieces with and without alternating bending for A)

DC04 and B) AA5182 [148]

Based on the measured springback angle for both materials, it can be concluded that the

amount of springback reduction through alternating bending for DC04 and AA5182 is

approximately 23% and 32%, respectively. Furthermore, the results reveal that the springback

of the aluminium alloy is higher than DC04 because of its smaller value of YOUNG modulus. To

investigate the effects of kinematic hardening as a result of alternating bending on the

springback of workpieces, numerical simulations were performed. The numerical results

based on FEM-simulation for draw bending of the U-Channel with macro-structured tools

confirm that the retention force caused by alternating bending in the flange area compensates

for the springback of the workpiece in both hardening models. As Figure 7-7 shows, the

springback will be reduced by generating alternating bending for pure isotropic, pure

kinematic, and also combined hardening types in both testing materials. Based on the results

of the simulations, the amount of springback by the pure isotropic hardening model is slightly

higher than the pure kinematic model. This is due to the larger level of the stored energy

density predicted by the simulation with the isotropic hardening model [173]. Furthermore,

the springback reduction through alternating bending in the pure kinematic hardening model

is more than the pure isotropic model (see the values below the diagram). This is because of

the induced BAUSCHINGER effect as a result of alternating bending. In other words, the

BAUSCHINGER effect has an additional influence on the compensation of springback. In

summary, the retention force induced through alternating bending reduces the amount of

springback during deep drawing with macro-structured tools. Moreover, the BAUSCHINGER

effect because of alternating bending can reduce further the springback behaviour of materials

with predominately a kinematic hardening effect.

DC04A)

δ = 0.2 mm

δ = 0.0 mm

AA5182B)

δ = 0.2 mm

δ = 0.0 mm

Results

69

Figure 7-7: Springback angle of U-Channels under different hardening types for A) DC04 and B)

AA5182

7.2 VERIFICATION OF THE DEVELOPED ANALYTICAL METHODS

In this section, the developed analytical methods to describe the process, as well as the

criteria for the prediction of process limits and dimensional accuracy of the parts, are verified

through experimental tests and numerical simulations.

7.2.1 Verification of the developed process model: forming energy

To verify the analytical model for calculation of the total energy for deep drawing of rotationally

symmetric geometries, the analytical results are compared with FEM results and experimental

tests. In these test series, samples from DC04 with two different initial outer radius of r0 = 90

and 100 mm, two different sheet thicknesses of s0 = 1.0 and 0.6 mm under two different

immersion depths of δ = 0.2 and 0.04 are considered. Based on the experimental draw-bend

test, the friction coefficient between the tools and the workpiece is set to be µ = 0.15 for both

analytical and numerical calculations. The experimental tests are repeated five times in order

to get reliable information about the statistical repeatability of the results. Figure 7-8 compares

the calculated total energy Et from Equation 6-6 with measured values from experimental tests

and also FEM results. The diagrams reveal that the analytical and numerical results exhibit a

very good agreement with the experimentally determined for forming energy in the whole

matrix of the study. The error indicators in both diagrams show a very small deviation from

the five repetitions per parameter set in the experiments.

Immersion depth δ = 0.0 mm Immersion depth δ = 0.2 mm

DC04A)

Springback

reduction in %28 19 23

Sp

rin

gb

ack a

ng

le β

1in

°

Iso.Kin. Iso-Kin Exp.0

4

8

12

16

20

23

AA5182B)

Sp

rin

gb

ack a

ng

le β

1in

°

0

4

8

12

16

20

Iso.Kin. Iso-Kin Exp.

Springback

reduction in %31 21 26 23

x

y β1

Results

70

Figure 7-8: Comparing the analytical calculated total forming energy with experimental and FEM

results for deep drawing of rotationally symmetric cups with outer radius of A) 90 mm and B) 100

mm [130]

The results verify the model for the calculation of total energy. However, as depicted in Figure

7-8-B, a bottom crack occurs in deep drawing of the part with an initial outer radius of

r0 = 100 mm, a thickness of s0 = 0.6 mm, and an immersion depth of δ = 0.4 mm. In order to

verify the applicability of the analytical model developed for more complex geometries by

means of the assumptions mentioned in section 6.3, the rectangular macro-structured tools

are used. In order to use the energy method for the rectangular cups, it should be divided into

four rotationally symmetric corners and four straight sections. In the straight sections there is

no compressive tangential stress and as a result the ideal forming energy Eid will be eliminated

from the total forming energy Et. For the other equations, the tool radius will be assumed to

be infinite. With these simplified assumptions, the total energy can be calculated. As the

diagram in Figure 7-9 shows, there is also a very good agreement between the calculated

energy and the experimentally and numerically determined values.

Figure 7-9: Comparing the analytically calculated total forming energy with experimental and FEM

results for the deep drawing of rectangular cups

Fo

rmin

g e

ne

rgy in

kJ

s0 = 0.6

δ = 0.2

0

2

4

6

8

s0 = 0.6

δ = 0.4

s0 = 1.0

δ = 0.2

s0 = 1.0

δ = 0.4

Fo

rmin

g e

ne

rgy in

kJ

0

2

4

6

8

s0 = 0.6

δ = 0.2

s0 = 0.6

δ = 0.4

s0 = 1.0

δ = 0.2

s0 = 1.0

δ = 0.4

Outer radius r0 = 90 mmA) Outer radius r0 = 100 mmB)

Experimental Analytical FEM

Cra

ck

Cra

ck

0.0

0.4

0.8

1.2

1.6

2.0

Fo

rmin

g e

ne

rgy in

kJ

s0 = 0.6

δ = 0.2

s0 = 0.6

δ = 0.4

s0 = 1.0

δ = 0.2

s0 = 1.0

δ = 0.4

Drawing depth: 30 mm

Experimental Analytical FEM

85 mm

205 mm

160 mm

35

mm

Results

71

However, the diagram shows the analytical determined energies are always slightly lower

than the experimentally determined energy since the in-plane shearing in the transition area

between the corner areas and the straight regions is not considered in the calculations.

Furthermore, small unconsidered parts of the sheet metal between the corners and the

straight parts (dark parts of the sheet metal in Figure 7-9) are not subjected to the calculation

because of their indefinable geometry. These results show that with the ability to calculate all

the individual energy terms for deep drawing with macro-structured tools, it is possible to

design the most energy-efficient process regarding the process input parameters.

7.2.2 Verification of the developed criterion: prediction of wrinkling

In order to control the accuracy of the criterion developed for the prediction of wrinkling,

experimental deep drawing processes with macro-structured tools were performed. For this

end, macro-structured tools with a constant wavelength of λ = 8.0 mm were used. Since the

thickness plays an important role on the buckling stiffness of the sheet metal, and thereby the

critical tangential stress σt,cr for wrinkling, blanks with an initial outer radius of r0 = 100 mm in

three different thicknesses of s0 = 0.3, 0.5 and 0.6 mm were subjected to the tests. In the

following, the critical stress σt,cr for wrinkling (see Equation 6-14) will be compared with the

present tangential stress on the free end part of the sheet metal (Equation 6-2). As mentioned

in the section on the Development of a criterion for the prediction of wrinkling, for a stable

process, the tangential stress σt must always be below the critical tangential stress σt,cr. This

criterion is controlled in this section. Figure 7-10 shows the results of the experimental tests

for the verification of the criterion developed regarding the prediction of wrinkling. Here all the

tests were repeated five times in order to get adequate information about the statistical

repeatability of the results. The results of the tests with a sheet thickness of s0 = 0.6 mm are

depicted in Figure 7-10-A. Here, the progress of the tangential stress at the free end part of

the sheet (most unstable part of the flange area, see section 6.1.1) over the punch

displacement is shown with a green line. The critical tangential stress for the onset of buckling

σt,cr, which is shown with a red line is almost 480 MPa. As the figure shows, the present

tangential stress on the free end part of the flange is always below the critical stress.

Therefore, there is no wrinkling expected in this case. The experimentally deep drawn cup

reveals that there is no wrinkling up to a punch displacement of h2 = 45 mm. However,

repeating the test with a thinner sheet metal (s0 = 0.3 mm) leads to dropping off of the critical

tangential stress σt,cr up to 120 MPa. As Figure 7-10-B shows, from the beginning of the

process (by punch displacement of h1 = 35 mm), the present tangential stress of the sheet

metal is always above an allowable value in the uncertain zone. As a result, it is expected that

wrinkling will be initiated at the beginning of the process. The experimentally deep drawn part

shown in Figure 7-10-B verifies this prediction. However, the critical situation should be for

the sample with a sheet thickness of s0 = 0.5 mm. As the Figure 7-10-C shows, at the

beginning of the process, the tangential stress at the free end part of the sheet is below the

critical stress (σt,cr = 310 MPa), and therefore no wrinkling should be expected at the

beginning. Continuing the process, the tangential stress reaches the critical value and enters

Results

72

into the uncertain zone by a punch displacement of 43 mm. It means that the wrinkling should

be initiated at the free end part of the sheet. This statement is verified through the experiment.

As shown on the figure, by a punch displacement of h2 = 45 mm, small wrinkling can be seen

on the free end part of the sheet. The experimental results show a good agreement with the

analytical criterion for the prediction of wrinkling in the deep drawing process with a macro-

structured tool. In the next section, the validity of the criterion developed for the prediction of

bottom cracks will be examined.

Figure 7-10: Prediction of wrinkling for samples with a sheet thickness of A) s0 = 0.6 mm,

B) s0 = 0.3 mm and C) s0 = 0.5 mm

7.2.3 Verification of the developed criterion: prediction of bottom

cracks

As discussed in section 6.4, for a stable process, the total forming force Ft (see Equation 6-4)

should be less than the critical bottom crack force Fbc (see Equation 6-22). In order to verify


0

200

400

600

800

1000

30 40 50 60

Tange

ntial str

ess in M

Pa

70

Wrinkling

Material: DC04

Outer radius: r0 = 100 mm

Inner radius: ri = 50 mm

Immersion depth: δ = 0.2 mm

Wavelength: λ = 8.0 mm

Sheet thickness: s0 = 0.6 mm

σt,cr from Equation 6-6

σt from Equation 6-2

A)

No wrinkling

Uncertain zone


0

200

400

600

800

1000

30 40 50 60

Tange

ntial str

ess in M

Pa

70

Wrinkling

h1 h2 No wrinkling

Uncertain zone

h1 h2


0

200

400

600

800

1000

30 40 50 60

Tange

ntial str

ess in M

Pa

70

Wrinkling

No wrinkling

Uncertain zone

h1 h2


Sheet thickness: s0 = 0.3 mmB)

Sheet thickness: s0 = 0.5 mmC)

Results

73

the criterion developed regarding the prediction of bottom cracks, the critical bottom crack

force for all the test series are calculated using the Equations 6-22 to 6-25. Figure 7-11

compares the measured maximum punch forces from the test series introduced in section

7.2.1 with the analytically calculated forces and also the corresponding bottom crack forces

Fbc.

Figure 7-11: Prediction of bottom cracks through the critical punch force for sheet metals with outer

radius of A) 90 mm and B) 100 mm

Figure 7-11-A shows that the forming force for the sample with an initial outer radius of

r0 = 90 mm is always below the critical force. Therefore, no bottom cracks are expected in

this test series while the experimental tests verify this prediction. However, as shown in

Figure 7-11-B, for the samples with an initial outer radius of r0 = 100 mm, the critical punch

forces from Equation 6-22 are always slightly higher than the analytically calculated and

experimentally measured forces, except for the sample with a thickness of s0 = 0.6 mm and

an immersion depth of δ = 0.4 mm. Therefore, bottom cracks can be expected in this case

and the corresponding experiment verifies this prognosis. These results approve the adequacy

of the criterion for the prediction of bottom cracks in the deep drawing process with macro-

structured tools.

7.2.4 Verification of the developed criterion: prediction of

dimensional accuracy

To control the influence of alternating bending on the residual stress of the parts during the

deep drawing process with a macro-structured tool, this process should be compared with a

conventional process. Furthermore, in order to take the dependency of the material into

account, the mild steel DC04, as well as the aluminium alloy AA5182, were subjected to

investigation. To use the half-analytical method described in section 6.6, the processes were

Forc

e in k

N

s0 = 0.6

δ = 0.2

s0 = 0.6

δ = 0.4

s0 = 1.0

δ = 0.2

s0 = 1.0

δ = 0.4

0

20

40

60

80

100

120

Forc

e in k

Ns0 = 0.6

δ = 0.2

s0 = 0.6

δ = 0.4

s0 = 1.0

δ = 0.2

s0 = 1.0

δ = 0.4

0

20

40

60

80

100

120

Cra

ck

Experimental Analytical Bottom crack force Fbc


Results

74

simulated by means of FEM. In the simulation sets, the deep drawing process with

conventional tools was considered as a reference and is compared with the macro-structured

deep drawing process with two different immersion depths of δ = 0.2 and 0.4 mm,

respectively, and a constant wavelength of λ = 8.0 mm. The results of the FEM are based on

2D FEM-simulations with an isotropic hardening assumption. The workpiece consists of

eleven elements over its thickness with a maximum element size of 0.1 mm. The blanks with

an initial outer radius of r0 = 90 mm and a sheet thickness of s0 = 0.6 mm were chosen for

the analysis. As mentioned in previous sections, based on experimental determinations, the

friction coefficient was set to be µ = 0.15 for the simulations. The conventional processes are

simulated using a spacer ring with a height equal to the sheet thickness in order to realise a

uniform surface pressure in the flange area. The simulations consist of two steps: deep

drawing up to the predefined height and consequently unloading the workpiece to simulate

the springback of the parts. To evaluate the opening gap of the split rings from Equation 6-39,

it was necessary to determine the bending moment as a result of the residual stress 𝑀bres. For

this purpose, the progress of the tangential residual stress 𝜎tres over the sheet thickness on

the wall of the part (where a ring will be cut out), should be determined based on the FEM

results. Figure 7-12 shows these stresses at the height of 35 mm from the bottom of the

parts. Consequently, the opening gap from Equation 6-39 can be evaluated based on

numerical integration of the shown residual stress curves. In order to verify the results of the

developed criterion, the results should be compared with real values from the experimental

test. The experimentally deep drawn parts are cut out by means of an electrical discharge

machining (EDM) process to prevent the external residual stresses through the cutting

process, because it does not require high cutting forces for removal of the material.

Figure 7-12: Tangential residual stress over the sheet thickness in the cup made by conventional and

macro-structured processes from A) DC04 and B) AA5182.

Conventional Macro-structured: δ = 0.2 mm Macro-structured: δ = 0.4 mm

Wall thickness s0 in mm

0-0.5 0.5

Tange

ntial re

sid

ual str

ess

in M

Pa

0

300

-300

200

100

-100

-200

Inner surface Outer surface

400

-400

0-0.5 0.5

Tange

ntial re

sid

ual str

ess

in M

Pa

0

300

-300

200

100

-100

-200

Inner surface Outer surface

400

-400

Wall thickness s0 in mm

DC04A) AA5182B)

s0

z

r

35

mm

s0

z

r

35

mm

Results

75

The opening gap of each ring after splitting was measured and is listed in Table 7 2. Moreover,

the calculated values for the opening gap, based on the half-analytical method, are also

presented in the same table in order to compare with the experimental results. The

experimental results exhibit a very good agreement with the half-analytically determined

opening gap of the split rings. The results reveal that regardless of the material, by increasing

the immersion depth in the macro-structured deep drawing process, the amount of residual

stresses can be reduced. It is because of the superimposed tensile stress as a result of

alternating bending, which is transferred from the punch to the material. This superimposed

tensile stress changes the proportion of the residual tensile and compression stress through

the sheet thickness. As a result, the stored elastic energy and consequently the springback of

the rings after cutting will be decreased. The same effect is also seen in [174].

Table 7-2: The opening gap of split rings based on experimental tests and calculated values.

Material

Opening gap in mm

(conventional)

Opening gap in mm (macro-structured)

δ = 0.2 mm δ = 0.4 mm

Measured Calculated Measured Calculated Measured Calculated

DC04 50 48.7 45 44.1 37 36.9

AA5182 95 99.2 73 81.1 64 71.8

In other words, the created non-frictional retention force can be used positively to compensate

the springback behaviour of the workpiece. Furthermore, the additional plastic deformation

due to the alternating bending decreases the elastic strains. This can also lead to a reduction

of springback in the deep drawn parts with macro-structured tools. Figure 7-13 shows the top

view of the split rings and gives an overview regarding the effects of the material and the

process on residual stress in the deep drawing process.

Figure 7-13: Results of the ring splitting test for cups made by conventional and macro-structured

tools from A) DC04 and B) AA5182.

DC04A)

δ = 0.2 mm

δ = 0.4 mm

Conventional

AA5182B)

δ = 0.4 mm Blank initial radius: r0 = 90 mm

Die inner radius: ri = 50 mm


Wavelength: λ = 8.0 mm

δ = 0.2 mm

Conventional

Results

76


Within the scope of this chapter, the analytical model and criteria developed in the previous

chapter were verified by means of the FEM, as well as experimental tests. Regarding process

stability, it was confirmed that the onset of wrinkling can be predicted analytically based on

the in-plane buckling theory. It was shown that as long as tangential stress at the free end

part of the sheet is below the critical tangential stress, no wrinkling is expected in the process.

The results verified also that the developed criterion can predict the process stability regarding

bottom cracks based on the energy method. For this purpose, the maximum allowable punch

force is a good benchmark for comparison with the forming force.

Moreover, there was a good agreement between the experimentally measured and

analytically calculated values, regarding the residual stresses in the workpiece. It was also

shown that the residual stresses in the deep drawing process with macro-structured tools are

generally smaller than by the conventional process.

Transfer of the methodology into the complex geometries

77

8. TRANSFER OF THE METHODOLOGY INTO THE

COMPLEX GEOMETRIES

Deep drawing of complex geometries is widely used in sheet metal forming processes to

produce irregularly shaped components. The formed asymmetric cups are needed in a variety

of industrial uses. It is required for automotive applications and aerospace parts, as well as a

wealth of other products. Rectangular and elliptic cups with large aspect ratios are used for

electrical parts such as battery containers, semi-conductor cases, and crystal vibrators. In this

chapter, the transferability of the developed approach into complex geometries is

investigated.

8.1 CONSTRUCTION OF A DEEP DRAWING TOOL TO FORM A T-CUP

In order to control the transferability of the already proposed technology into complex

geometries for industrial applications, a modular deep drawing tool with a relatively complex

geometry, a so called “T-Cup”, was constructed in the scope of this thesis. Figure 8-1 shows

a detailed view of the tool. The form entails concave and convex shapes, as well as straight

parts to have a variety of stress states in the part. The modularity of the tool allows all

functional surfaces to be changed, like the die edge radius and the flange area, regarding their

forms and dimensions. The modular die edge from the cold work tool steel 1.2379, with a

radius of 10 mm, was hardened up to 60 HRC and afterwards grinded and polished to reach

a surface roughness of Ra = 0.1 and Rz = 3.2 μm. The segments of the flange area also from

tool steel 1.2379, with a delivered hardness of 25 HRC, were milled to fabricate the macro-

structures with a constant wavelength of λ = 10 mm, based on preliminary investigations by

means of the FE and the analytical method. These segments were also ground and polished

manually to reach a surface roughness of Ra = 0.6 and Rz = 6.5 μm. Here, polishing the macro-

structures using a classical polishing machine was not applicable. These parts were

assembled from several segmented pieces on the fixing plate. To compare the macro-

structured deep drawing with the conventional process, conventional flange segments with

the same inner and outer geometries were also constructed and manufactured. These

segments have the same hardness and surface roughness as the die edge radius. The surface

properties of both conventional and macro-structured tools are listed in Table 8-1.

Table 8-1: Surface-related properties of the modular deep drawing tool

Part Hardness (HRC) Ra (μm) Rz (μm)

Die edge radius 62 0.1 3.2

Conventional flange 62 0.1 3.2

Macro-structured flange 25 0.6 6.5


78

The differences between the roughness and the hardness values were because of their

manufacturing processes. To apply the blankholder force, six pneumatic springs were used.

The nitrogen medium (N2) used in each spring, with the maximum filling pressure of 15 MPa,

realises a quasi-steady working spring stiffness of 625 N/mm. The 80 mm nominal stroke of

springs determines the traveling distance of the tool and also its maximum drawing depth.

The tool is cleaned with a proper cleaning agent before being used to ensure a total lubricant-

free deep drawing process. The hydraulic press machine RZP 250 from Röcher Maschinenbau

(see section 5.1.2) was used to form the T-Cups because of its high power performance and

room capacity.

Figure 8-1: Detailed view and components of the T-Cup deep drawing tool

Pneumatic spring

Punch

Base/ punch plate

Die plate

Guide pillar

Bla

nkh

old

er p

late

Spacer

50 mm

50 mm

300 mm

600 mm

47

5 m

m

Fixing plate

Segmented flange

Segmented die

edge radius


79

8.2 DETERMINATION AND EVALUATION OF OPTIMUM INITIAL BLANK

GEOMETRY

Deep drawing of complex parts requires several intermediate steps to avoid any defects and

to achieve the final desired geometry successfully. Using the optimum initial blank geometry

has many advantages in the deep drawing process. The optimum initial blank geometry not

only reduces the material costs of the produced part, but also improves the deep drawing

quality, thickness distribution and formability of the part, and minimises forming defects.

However, finding the optimum blank geometry could be difficult and time consuming. Since

experimental trial-and-error methods for achieving the initial blank geometry are time

consuming and expensive, other methods were sought for optimum blank design in the sheet

metal forming operations. Different approaches for blank geometry design have been reported

in the literature [175]. These methods can be classified as the slip-line field method [176],

geometrical mapping [177], the analogy method [178], ideal forming [179], the inverse

approach [180], backward tracing [181], and the sensitivity analysis method [182]. The blank

shape design in this thesis was carried out using the slip-line field method. Slip-line field theory

is based on the analysis of a deformation field that is both geometrically self-consistent and

statically admissible. Slip lines are planes of maximum shear stress and are therefore oriented

at 45° to the axes of principal stress [183]. Using this graphical-analytical approximation

method, the initial blank geometry was determined for deep drawing of the T-Cup as shown

in Figure 8-2.

Figure 8-2: Determination of the optimum initial blank geometry using Slip-line field theory

To prognosticate the process stability regarding bottom cracking and wrinkling, the analytical

method which was introduced in section 6.3 can be applied for the T-Cup. For this purpose,

the sheet metal determined can be divided into several geometrical analysable parts with

definable dimensions as shown in Figure 8-3. Through superposing the forming force of all

275 m

m

535 mm430 m

m

400 mm


80

parts, it is possible to find out the required punch force. Subsequently, it can be compared

with the bottom crack force Fbc and used to predict the process stability regarding bottom

cracks. Moreover, considering the stress states in each part which are labelled with directional

arrows in Figure 8-3, it is possible to specify the most unstable areas regarding wrinkling. Parts

2, 4 and 6, with a quarter circle geometry and 100 mm radius, undergo tensile-compressive

stress with superimposed alternating bending. Because of the existing tangential compressive

stress in these areas, these parts are the most probable wrinkling zones of the sheet metal.

Therefore, the model for analysing the rotationally symmetric deep drawing process can be

applied to these parts. Straight parts 1, 5 and 7 sustain alternating bending with superimposed

tensile stress.

Figure 8-3: The superposing method for analysing the deep drawing of the T-Cup

Hence, the corresponding forming force can be calculated by excluding the ideal forming force

from the punch force calculation of the rotationally symmetric parts. In part 3, a tensile-tensile

stress state leads to stretch forming conditions. The stretch of the sheet with a thickness of

s0 = 1.0 mm can be estimated by the formulation of OEHLER und KAISER [184], which is verified

by LANGE in [166] as following:

𝐹sf =𝐴1𝜂F𝜎ym ln

𝐴1𝐴0

8-1

Here, A0 is the original area of the sheet, A1 is the increased area after stretch drawing, and

σym is the mean yield stress. The forming efficiency is x = 0.5 to 0.7. If the stresses are

distributed uniformly over the entire surface, this factor should be ηF = 0.7, and for unequal

load distribution the forming efficiency should be ηF = 0.5 [166].

50

150

25

76

5

4

3

2 1

30


81

8.3 TEST PROCEDURE AND INTERPRETATION OF RESULTS

In order to investigate the usability of macro-structured tools, as well as verification of the

analytical model for a T-Cup, experimental tests were performed. For this purpose, sheet

metals from DC04 with 1.0 mm thickness were cut out through the laser-beam cutting

method, corresponding to the already determined optimum shape. The sheet metals were

cleaned after deburring with acetone to remove the initial corrosion-preventing oil on the sheet

metal. Subsequently, the cleaned and deburred sheet metals were subjected to the deep

drawing process with macro-structured tools with an immersion depth of δ = 0.2 mm. The

exact immersion depth can be adjusted by using a contouring spacer ring. Figure 8-4 shows

the progress of the drawing process in three stages. As the Figure shows, there is a stable

process regarding both the failure cases of bottom cracking and wrinkling.

Figure 8-4: Progress of the deep drawing of the T-Cup with macro-structured tools

In order to make a statement about the accuracy of the analytical criterion to predict the

occurrence of bottom cracks, the required punch force to form the T-Cup, calculated from the

analytical model, was compared with the FEM and experimental results. The process was

simulated in 3D with the FEM. For the simulation, solid shell elements with 2 mm element

size which consist of 5 integration layers (1 integration point on each layer) were used. To

increase the accuracy of the simulation, the anisotropic yield function of HILL was applied,

with the corresponding r-values determined from uniaxial tensile tests. The frictional force

based on the COULOMB model with a friction coefficient of μ = 0.15 for the whole tool was

considered. Figure 8-5 shows the load-displacement curve for deep drawing of the T-Cup,

regarding the experimental, FEM and analytical results. The results show that the progression

of forces over the punch displacement tended to be similar. However, for determination of

bottom crack by means of developed criterion only the maximum value of the punch force is

relevant, therefore this is considered for the calculations.

Sheet in initial stage:

Drawing depth: 0 mm

Sheet in middle stage:


Sheet in final stage:


100 mm100 mm100 mm


82

Figure 8-5: Load-displacement curve for the deep drawing of the T-Cup

To quantify the punch force and also to be able to assess the process window regarding

bottom cracks, the critical bottom crack force from Equation 6-22 was calculated. All results

regarding the maximum punch force from the experimental tests, FEM calculations and the

analytical model are summarised in Figure 8-6 and compared with the relating bottom crack

force Fbc.

Figure 8-6: Comparison between the maximum punch force from the experimental tests, FEM

calculations and the analytical model

In the experimental tests, the average punch force was 205 kN, for the deep drawing of the

T-Cup from DC04 with a 1.0 mm thickness, 10 mm wavelength and 0.2 mm immersion depth.

Comparing this value with the corresponding result from the FEM simulation shows that there

is approximately only a 7% difference. This good agreement, between the numerical results

and the experimental tests, reveals the accuracy of the applied material model and the

boundary conditions. However, the analytical model predicts a punch force of 180 kN, which

is 12% and 18% less than the experimental and FEM results, respectively. These deviations

from the real measured values and the FEM-based calculated punch force are because of

Material: DC04



Wavelength: λ = 10 mm

Experiment

FEM

Analytical0

50

100

150

200

250

0 10 20 30 40 50 60 70

Pu

nch f

orc

e in

kN


Maxim

um

punch

forc

e in k

N

0

50

100

150

200

250

300

Experiment AnalyticalFEM

350 Drawing depth: 70 mm

Material: DC04

Bottom crack force, Fbc




No bottom cracks

100 mm


83

ignoring the in-plane shearing in the transition area between the corner areas and the straight

regions. Moreover, the small parts of the sheet metal between the geometrically definable

segments were not considered in the calculations (see Figure 8-3). Comparing the punch force

from the analytical model with the bottom crack force (Fbc = 285 kN, see the dashed line in

Figure 8-6), it can be noted that no bottom cracks was expected. Obviously this statement

was verified with the experimental tests.

To investigate the stability of the process regarding wrinkling, the analytical criterion based on

buckling analysis which was proposed in section 6.4 should be used for the corners with high

probability of wrinkling (the green marked parts of the sheet metal in Figure 8-7). As shown in

Figure 8-7, the corners approaching a quarter circle and an outer radius of 100 mm were

subjected to the analysis. Setting the corresponding parameter into Equation 6-13, results in

the critical tangential stress of σt. cr = 721MPa, which is well over the acting tangential stress.

This consideration reveals that the analytical model expects no wrinkling for the deep drawing

of the T-Cup, which was verified with the experimental tests.

Figure 8-7: Buckling analysis to predict the process stability regarding wrinkling for the deep drawing

of the T-Cup

These experimental results are in good agreement with the prediction of the analytical criteria.

However, in order to compare the total punch force resulting from deep drawing with macro-

structured tools with the conventional process, some further tests were carried out. Here, the

conventional tools were used which were introduced in section 8.1. In order to ensure the

comparability of results, the same blank geometries were used for deep drawing with macro-

structured tools as for the conventional process. Here, the conventional deep drawing

processes were done with use of a spacer ring, as well as the direct blankholder force. The

tests were performed initially in a lubricant-free and subsequently in lubricated conditions. The


200

300

400

500

30 40 50 60

Str

ess in

MP

a

70

600

700

800

No wrinkling R50

Outer radius: r0 = 100 mm

Inner radius: ri = 50 mm




σt,cr from Equation 6-6

σt from Equation 6-2

Uncertain zone


84

corresponding results are presented in Figure 8-8-A, and compared with results of the deep

drawing process with macro-structured tools in lubricant-free conditions. The results show

that for deep drawing of a T-Cup with conventional tools in lubricant-free conditions with use

of space ring (with the height equal to the sheet thickness), the total punch force amounts to

almost 220 kN. This is approximately 10% more than deep drawing with macro-structured

tools. Here, it must also be taken into account that according to Table 8-1, the conventional

tools have much better surface roughness as well as hardness, which play a very significant

role on the tribological behaviour of the tools. However, repeating the test without a spacer

ring and direct application of blankholder force leads to bottom cracks in the workpiece. The

blankholder force which is provided from pneumatic springs varies during the test regarding

their spring stiffness. Figure 8-8-B shows the change of the blankholder force and resulting

surface pressure as a function of drawing depth. Changing the test conditions through

lubricating the sheet metal with WISURA ZO 3368 leads to a reduction of the punch force of

up to 195 kN in the deep drawing process with conventional tools and use of a space ring.

This value is very close to the required punch force for the deep drawing of a T-Cup with

macro-structured tools in lubricant-free conditions. Therefore, it proves that the macro-

structured tools can fulfil the function of lubricants regarding the reduction of friction to realise

a lubricant-free forming process even for complex geometries. Moreover, the results show

that no bottom cracks occur when applying the blankholder force for the conventional process

in lubricated conditions. However, the punch force increases up to 225 kN, which is

approximately 12% higher than a macro-structured tool.

Figure 8-8: Comparison between the conventional and macro-structured deep drawing processes; A)

Difference between the conventional and macro-structured deep drawing processes regarding the

total punch force and B) the progress of the blankholder force as a function of drawing depth

Material: DC04 Sheet thickness: s0 = 1.0 mm Drawing depth: 70 mm Lubricant: WISURA ZO 3368

A) B)

Maxim

um

pu

nch

fo

rce

in

kN

0

50

100

150

200

250

300

350

Conventional

Sp

ace

r ri

ng

Lubricant-free

Bla

nkh

old

er,

cra

ck

Bla

nkh

old

er

Lubricated

Sp

ace

r ri

ng

Lubricant-free

δ=

0.1

mm

Macro-structured

Bla

nkh

old

er

forc

e in

kN

Drawing depth in mm

450

630

720

540

Su

rface

pre

ssu

re in

MP

a

4.8

7.8

5.8

6.8

10 30 50 70

Blankholder force

Surface pressure


85

In order to clarify the advantages of the macro-structured process over the conventional

process, the parts which are formed in lubricant-free conditions are compared in Figure 8-9.

As mentioned above, there is no significant difference between the forming force in macro-

structured and conventional tools using the spacer ring in lubricant-free conditions

(approximately 10%). However, as shown in Figure 8-9-A, the bottom area of the deep drawn

part made by a conventional tool is not formed plastically, and therefore there is a bending

fold in the corner of the part. Despite this, in the deep drawn part made by macro-structured

tools, the bottom area is totally flat. This is because of the possibility to control the material

flow in deep drawing with macro-structured tools through alternating bending mechanisms.

Figure 8-9: Comparison between the part properties made by conventional and macro-structured

tools in lubricant-free conditions; A) Conventional tool and B) macro-structured


Several structural parts of automobile components have complicated and very often

asymmetrical geometries, which require their production to be performed by means of the

deep drawing process. This fact indicates the importance of technology transfer into the

industrial parts. To control the transferability of the proposed approach and also the suitability

of the criteria developed to predict the process limits, a tool for deep drawing of the T-Cup

was constructed. The flange area of the tool is macro-structured, based on an already

developed model to have a stable process. The advantage of the T-Cup is because of its variety

of stress states. Slip-line field theory was used to determine the optimum initial blank

geometry. Firstly, the experimentally deep drawn parts with macro-structured tools in

lubricant-free conditions had neither wrinkles nor bottom cracks. For verification of the model

with the experimental observations, the required forming force for the T-Cup was determined

analytically and numerically. These forces were in a very good agreement with the

experimentally measured force. Therefore, it can be concluded that the model is able to

predict the required forming force of the complex geometries as well. Comparing this force

with the critical bottom crack force, showed that the process is in a safe working area and no

bottom cracks are expected. The process was also analysed regarding wrinkling to control the

Conventional process

with blankholder force

Conventional process

with spacer ring

ConventionalA) Macro-structuredB)


86

accuracy of the criterion for the prediction of wrinkling. The implemented buckling analysis

showed that the critical tangential stress for the initiation of wrinkling is over the true

compressive stress in corners of the part. Therefore, the analytically developed criterion

predicted no wrinkling in the specimen and it was proven with real experimental tests.

Furthermore, it was shown that the macro-structured deep drawing process in lubricant-free

conditions requires less punch force compared to the conventional process in lubricated

conditions through application of the direct blankholder force. Moreover, the results showed

that using a spacer ring in the deep drawing process with conventional tools in lubricated

conditions requires almost the same amount of punch force as the macro-structured process.

The results revealed that macro-structured tools can be completely replaced with conventional

tools to reduce the frictional force and realise a lubricant-free forming process. However, the

results showed that there is no significant difference in the punch force between conventional

tools using a spacer ring and macro-structured tools in lubricant-free conditions. But, in deep

drawing with a macro-structured tool the material flow can be controlled, in a way which can

influence the material strain in the bottom part of the T-Cup. Thereby, unlike conventional

process, the material in this area can be formed plastically and consequently no bending folds

occur in this area.

Possibilities for process improvement

87

9. POSSIBILITIES FOR PROCESS IMPROVEMENT

It has already been shown that to eliminate the lubrication from the deep drawing process

macro-structured tools can be used. Using macro-structured tools can reduce the amount of

frictional forces in the flange area. As a sub-target, it is also possible to enlarge the process

window compared with the conventional process. For this end, the total punch force of the

lubricant-free deep drawing process with macro-structured tools should be less than the

punch force of the conventional process in lubricated conditions. Therefore, in order to

improve the process regarding the enlargement of the process window, a further reduction of

the punch forces is necessary. Considering the energy component for deep drawing with

macro-structured tools as introduced in section 6.2, the ideal forming energy and the energy

for bending over the die edge radius are unchangeable for a particular process. However, a

further reduction of friction on the flange area, as well as reduction of the alternating bending

energy, can be considered as two improving measures for the enlargement of the process

window in deep drawing with macro-structured tools. In this chapter, these measures are

introduced and their influences on the reduction of punch force are examined through the

FEM and experimental tests.

9.1 USE OF ROTARY ELEMENTS AS MACRO-STRUCTURED ELEMENTS

For further reduction of frictional forces, spherical elements with very small friction

coefficients should be combined with line structures in the flange area of the deep drawing

tools to induce alternating bending. For this purpose, rotary spherical elements (like ball

casters) can be used [185]. By using these elements, the contact lines can be reduced to

contact points, minimising the contact area even further. This leads to a reduction of the

frictional force and changes the type of friction from sliding to rolling. However, using the

spherical elements in the flange area reduces the supporting points against wrinkling.

Therefore, the spherical elements should only be applied in uncritical areas regarding

wrinkling, like the straight parts of a deep drawing part.

Based on the positive results achieved in Chapter 7, the amount of friction reduction for an

oval-form macro-structured tool with combined forming elements compared to a conventional

deep drawing tool (see Figure 9-1-A) was examined through numerical investigation [114]. To

determine the non-frictional forming force, which means without any friction, a third simulation

was carried out. As Figure 9-1-B shows, approximately 70 kN were required for the forming

process in the idealised case without friction. Therefore, this result can be considered as the

lowest forming force for deep drawing of this part. As the diagram shows, a punch force of

155 kN is required for the deep drawing process with a conventional tool, and 85 kN are solely

necessary to overcome the frictional forces caused by using a sliding friction coefficient of

µ = 0.15. Therefore, 55% of the total punch force is due to the frictional force. Simulation

results from deep drawing processes with macro-structured tools combined with spherical

elements (with a friction coefficient of µ = 0.02 based on [186]) and line structures (with a


88

sliding friction coefficient of µ = 0.15) indicate a total punch force of 90 kN, which is equivalent

to a reduction of 42% compared to the conventional process. In this case of macro-structured

tools, about 10 kN are required to generate the alternating bending, but the frictional forces

account for only 10 kN. This means a reduction of 88% compared to the conventional process.

Comparing the macro-structured deep drawing process with a frictionless deep drawing

process shows that the punch force increases by only 22% because of alternating bending

and frictional forces. This can be used to enlarge the process window significantly.

Figure 9-1: Using rotary elements as macro-structured elements; A) Schematic of conventional and

macro-structured deep drawing tools, B) comparison between the conventional and macro-

structured deep drawing process regarding the punch force [114]

By using the industrial ball casters available on the market as forming elements, some other

advantages can be expected in terms of the process design and tool technology. Due to the

locally variable adjustability of the position and the height of the elements relative to the sheet

metal, a high level of flexibility can be achieved for setting the values of the wavelengths and

immersion depths. The ball casters are available in numerous different designs with regard to

the types of construction, diameter and permissible load suspension.

9.2 MACRO-STRUCTURED TOOLS WITH VARIABLE WAVELENGTHS

In this section, the already developed approach for the lubricant-free deep drawing process

can be improved regarding the locally adapted wavelengths in order to further reduce the

punch force and also the enlargement of the process window. The improving approach

considers all requirements of a lubricant-free deep drawing process for the broadest possible

material spectrum, based on macro-structured elements with variable wavelengths. So far, a

constant wavelength λ has been used for the macro-structuring, while the value of λ at the

most critical part of the flange is determined for a stable process regarding wrinkling. As

shown in section 6.1.1, this corresponds to the area of the greatest tangential compressive

A)

Pu

nch f

orc

ein

kN

0

40

80

120

160

Conventional

with friction

Macro-

structured

conventional

without friction

88

%

22

%

42

%

Line

structures

Spherical

structures

20

0 m

m

100 m

m

Punch

Macro-structuredConventional

200 mm

B)Ideal

Friction

Alternating bending

Top view of conventional and

macro-structured tools for FEA


89

stress; in the rotationally symmetric case, this is the outer free end part of the blank. As a

result of material flow, this area moves in a radial direction, whereby the tangential

compressive stress increases continuously because of the strain hardening of the sheet

metal. This effect has less importance for the low-strength materials, but it can be taken into

account in the selection of the wavelength λ for the deep drawing of (super) high-strength

materials. Figure 9-2-A shows how the tangential stress at the free end part of the sheet metal

increases during the process. In other words, the risk of wrinkling at the free end part

increases during the process. In order to compensate for this effect, the wavelength should

be reduced in the direction of material flow. This way allows macro-structures to be used with

a higher wavelength at the outer part of the flange, where there is no particularly high risk of

wrinkling. Furthermore, in contrast to the previous consideration, the number of alternating

bending mechanisms in the flange area can also be reduced since the number of macro-

structures can be smaller. This strategy reduces the required energy to generate the

alternating bending mechanism, which results in a reduced punch force and thus an increased

process window for the lubricant-free deep drawing, especially for (super) high-strength

materials.

Figure 9-2: Macro-structured tool with variable wavelength; A) Increasing the amount of tangential

compressive stress as a result of strain hardening, and B) developing a concept for a macro-structured

tool with variable wavelength

In order to be able to take the above mentioned relationships into tool design, the already

developed criterion for the prediction of wrinkling should be extended to find the locally-

adapted wavelength for a stable process. The amplitude of the macro-structures that so far

were assumed to be constant should be varied in the direction of the material flow during the

deep drawing process. For this purpose, the tangential stress at each time step can be

Increase the risk of wrinkling

t1t2t3t4

Stress

+

-

λ2λ3λ4 λ1

Variable wavelength

Tange

ntial str

ess

t1 t2 t3 t4

σt,cr for variable wavelength

σt,cr for constant wavelength

Progress of σt at free end part

Punch displacement / Time

A) B)


90

considered as the critical tangential stress σt,cr as shown in Figure 9-2-B for the four time step.

Knowing the amount of critical tangential stress per each time step, the wavelength can be

calculated locally based on the inverse calculation of the already developed criteria (Equation

6-14). In order to investigate the effects of the proposed approach regarding the reduction of

forming energy and punch force, as well as the enlargement of the process window, new

rotationally symmetric macro-structured deep drawing tools with variable wavelengths from

the tooling material described in 5.2, were provided for the experimental tests. As it depicted

in Figure 9-3, the wavelength varies from 11 mm to 8 mm in a radial direction of material flow

based on the extended analytical approach which is discussed above. For experimental

verification of the approach, the same material DC04 was chosen. In order to quantify the

advantages of deep drawing with variable macro-structured tools, the same experimental test

series which were carried out with a constant wavelength tool (section 7.2.3), are repeated

with a new tool design.

Figure 9-3: Macro-structured deep drawing tool with variable wavelength

As mentioned above, the experimental test matrix used in section 7.2.1 is repeated with a

new macro-structured tool. Here the blanks, with outer radius of r0 = 90 and 100 mm and two

different sheet thicknesses of s0 = 0.6 and 1.0 mm, were subjected for the test series. The

immersion depth varies from δ = 0.2 to 0.4. The measured punch forces for deep drawing

with variable macro-structured tools are compared with the previous results and shown in

Figure 9-4. As the diagram shows, the punch forces are always slightly reduced by using the

macro-structured tools with variable wavelengths (11 mm < λ < 8 mm). Based on these results,

it can be concluded that the modified macro-structured tool can further reduce the punch force

and improve the efficiency of the lubricant-free deep drawing process. Furthermore, despite

macro-structured tools with a constant wavelength, no bottom cracks occurred when using

variable wavelength tools for deep drawing samples with a thickness of s0 = 0.6 and an

immersion depth of δ = 0.4 mm.

8.0 mm

8.4 mm

9.0 mm

11.0 mm

Ø103

Ø225


91

Figure 9-4: Comparison between macro-structured tools with constant and variable wavelengths

regarding the punch force and enlargement of the process window for sheet metals with outer radius

of A) 90 mm and B) 100 mm


Within the scope of this chapter, two different methods were introduced to improve the

process regarding reducing the forming energy which leads to enlargement of the process

window. It was shown that the rotary spherical element reduces the contact lines to contact

points and changes the sliding friction condition to a rolling condition. The FEM results showed

that the share of friction from the total punch force can be reduced up to 88%. However, this

method can be applied only in the uncritical parts of the tool regarding wrinkling, because of

the loss of supporting points.

As a second method, it is possible to adjust the macro-structures to the acting tangential

stress at the free end part of the sheet metal locally. Through this method, a macro-structure

with a variable wavelength can be plausible. Generally, the tangential compressive stress

increases continuously at the free end part of the sheet metal during the process due to the

strain hardening of the material, while the radial tensile stress is always zero. Therefore, the

risk of wrinkling increases during the process. Based on this investigation, locally adapted

macro-structures were developed which leads to reduction of the bending energy at the outer

part of the flange area, and as a result, reduction of the punch force. The experimental results

showed that the punch force can be slightly reduced through this measure.

Summarising these statements, it can be concluded that by combining the improvement

methods, the punch force can be further reduced for the lubricant-free deep drawing process

and the process window can be enlarged regarding bottom cracks. It should also be taken into

account that these methods have no negative influences on the process stability regarding

wrinkling.

Pun

ch f

orc

e in k

N

s0 = 0.6

δ = 0.2

Constant wavelength: λ = 8 mm Variable wavelength: 11 < λ < 8 mm

s0 = 0.6

δ = 0.4

s0 = 1.0

δ = 0.2

s0 = 1.0

δ = 0.4

0

20

40

60

80

100

120

s0 = 0.6

δ = 0.2

s0 = 0.6

δ = 0.4

s0 = 1.0

δ = 0.2

s0 = 1.0

δ = 0.4

0

20

40

60

80

100

120

Cra

ck

Pu

nch

forc

e in

kN


Summary and conclusions

92

10. SUMMARY AND CONCLUSIONS

Friction is one of the most restricting parameters in sheet metal forming operations. In the

deep drawing process, lubricants are applied with the aim of reducing the friction between

the tool and the sheet metal for protection of the semi-finished products against corrosion,

reduction of tool wear and also the enlargement of process window. However, from both

economic and ecological points of view, it is strongly recommended to remove the lubricants

within the deep drawing process. For the upcoming process steps after forming, such as

joining and coating processes, which are absolutely sensitive to contaminants and oil, it is

essential to clean the workpieces from lubricants. This is carried out in post-treatment

processes by use of degreasing agents, which are solvent-based and therefore

environmentally unfriendly and unhealthy. In addition to lubrication, there are currently a

number of ways to reduce friction, like improving the surface quality (hardness as well as

roughness), tool coating, and surface texturing. However, for a total lubricant-free process,

the frictional forces should be reduced even further. Realising a total lubricant-free deep

drawing process leads to:

a reduction of process steps in production,

a reduction of mineral oil needs and

establishing a green forming technology.

The goal of this thesis was to develop a new deep drawing tool for lubricant-free applications,

which ensures the process window in lubricant-free conditions. Since the largest contribution

of the drawing force is the friction in the flange area, this part of the tool should be adapted

for the new tool design. In order to decrease the amount of frictional force in this area for a

given friction coefficient, the integral of the contact pressure over the contact area has to be

reduced. To achieve that, this area is macro-structured, which has only line contacts and thus

a smaller contact area with the sheet metal. This can increase the risk of wrinkling in the

unsupported sheet metal areas, because the usually utilised blankholder force is not

applicable. To avoid this effect, the geometrical moment of inertia of the sheet should be

increased by the alternating bending mechanism of the material in the flange area through

immersing the blankholder slightly into the drawing die. Figure 10-1 summarises the sequence

of approaches to realise the lubricant-free deep drawing process.

Figure 10-1: Procedural methods for realisation of the lubricant-free deep drawing process

Lubricant-

free deep

drawing

Elimination

of lubricant

Risk of

bottom

cracks

Reduction

of contact

area

Risk of

wrinkling

Alternating

bending

Approach

Leads to

Macro-

structured

tools

Approach

Leads to

Approach

Leads to


93

Therefore, the developed process can, besides the reduction of the contact area and the

blankholder force, also increase the resistance of the sheet metal against wrinkling.

Furthermore, through adjusting the immersion depth, it is possible to control the material flow

and enlarge the process window. The induced alternating bending to stabilise the sheet metal

against wrinkling during deep drawing with macro-structured tools generates an additional

force in the drawing direction. The created non-frictional restraining force can be used

positively to compensate the springback behaviour of the workpiece. Generally, following

positive effects can be achieved by deep drawing with macro-structured tools:

reducing the contact area,

reducing the integral of the contact pressure over the contact area,

reducing the total punch force through minimising the frictional force,

saving the added-cleaning time of the lubricant, as well as its disposal costs through

elimination,

enlargement of the process window regarding bottom cracks through reducing the

total punch force,

increasing the process stability regarding wrinkling through inducing alternating

bending,

compensation of the springback effect because of the induced alternating bending as

a non-frictional restraining force,

possibility to control the additional strain hardening, as well as the strain softening

through alternating bending for a material with predominantly isotropic or kinematic

hardening behaviour,

possibility to control the material flow by adjusting the amount of the immersion depth

and

no die spotting is required for the new tool systems.

Since the process stability is dependent on the two input parameters, local immersion depth

and local wavelength, it is necessary for a time efficient tool design to have an analytical model

and criteria to predict the process stability in a fast and accurate way in advance. Therefore, a

deep drawing process with macro-structured tools was modelled analytically to predict bottom

cracks and wrinkling by means of the energy calculation method, as well as plastic buckling

analysis, respectively. The feasibility of the process and also the accuracy of the model were

verified through numerical simulations, as well as experimental tests. The results from the

analytical model were in a very good agreement with both the numerical and experimental

results.

In order to control the transferability of the proposed approach into a more complicated

geometry, macro-structured as well as conventional tools for the deep drawing of the T-Cup

were constructed within the scope of this thesis. Both criteria to predict the process limits

were used for deep drawing of the T-Cup with macro-structured tools. Based on these

predictions, neither bottom cracks nor wrinkling were expected during the process and the

experimental results verified these prognoses. Moreover, the results showed that the


94

maximum punch force for deep drawing with macro-structured tools in lubricant-free

conditions were very close to that of conventional tools in lubricated conditions. The results

showed that using conventional tools with a blankholder force in lubricant-free conditions

produces bottom cracks in the workpiece. However, using a spacer ring for deep drawing with

conventional tools in lubricant-free conditions leads to a stable process with no significant

difference in the punch force compared to macro-structured tools. Nevertheless, due to the

ability of macro-structured tools to control material flow during the process, no bending folds

occurred on the bottom part of the specimen.

Moreover, two additional approaches were introduced to reduce the punch force further. The

use of rotary elements as macro-structures changes the type of friction from sliding to rolling,

which can reduce the frictional force further and leads to enlargement of the process window.

Although this is a complicated approach in application, it can be considered as a vision for

future work. Besides that, it was shown that locally adapting the wavelength regarding the

tangential stress can lead to further reduction of the punch force.

Although, the design for macro-structured tool can be relatively more time- and cost-

consuming, because of its form and additional wear minimising surface treatments like coating

and micro-texturing, it provides several economic and ecological advantages in production.

Since the main task of macro-structuring is to reduce the amount of frictional forces, it can be

used in the presence of lubrication to perform a minimum-quantity lubrication. Moreover, the

newly developed tool design can in combination with a common lubricating system reduce

the frictional forces further and thereby lead to increased enlargement of the process window.

Zusammenfassung

95

II. ZUSAMMENFASSUNG

Reibung stellt einen der am stärksten einschränkenden Parameter in der Blechumformung

dar. Bei Tiefziehprozessen wird deshalb Schmierstoff eingesetzt, um die Reibung zwischen

Werkzeug und Blech zu verringern, Halbzeug gegen Korrosion zu schützen, den

Werkzeugverschleiß zu begrenzen und das Prozessfenster zu vergrößern. Aus ökonomischen

und geoökologischen Gesichtspunkten besteht die Herausforderung, auf den Einsatz von

Schmierstoffen verzichten zu können. So muss beispielsweise der Schmierstoff für die

Folgeoperationen nach dem Umformen, wie Fügen und Beschichten, die überwiegend sehr

sensibel bezüglich Verunreinigungen und Öl sind, unbedingt entfernt werden. Die Reinigung

wird mithilfe von Entfettungsmitteln durchgeführt, die Lösemittel enthalten und somit

umweltunfreundlich und gesundheitsschädlich sind. Neben der Schmierung gibt es derzeit

eine Reihe von Möglichkeiten, um die Reibung zu verringern, z. B. eine verbesserte

Oberflächenqualität (Härte sowie Rauheit), die Werkzeugbeschichtung und die

Oberflächenstrukturierung. Trotzdem muss die Reibung für einen schmierstofffreien Prozess

noch weiter reduziert werden. Die Realisierung eines total schmierstofffreien

Tiefziehprozesses ermöglicht

eine Reduzierung der Produktionsschritte,

eine Reduzierung des Mineralölbedarfs und

die Realisierung einer „Green Technology“.

Das Ziel dieser Arbeit war die Entwicklung eines neuen Tiefziehwerkzeuges für

schmierstofffreie Anwendungen, welches die Stabilität des Prozessfensters sicherstellt. Da

die Reibung im Flanschbereich den größten Anteil an der Umformkraft hat, sollte dieser Teil

des Werkzeuges bei der neuen Werkzeugauslegung angepasst werden. Bei gegebener

Reibzahl muss das Integral der Flächenpressung über die Kontaktfläche reduziert werden, um

die Reibkräfte zu reduzieren. Durch die Makrostrukturierung reduziert sich die Kontaktfläche

zu einem linien- oder punktförmigen Kontakt. Dies kann zur Erhöhung der Neigung zur

Faltenbildung im Flanschbereich in den nicht gestützten Bereichen führen. Allerdings

ermöglicht das geringfügige Eintauchen des Niederhalters in die Struktur der Matrize eine

Erhöhung des Flächenträgheitsmoments des Bleches durch die eingebrachte Wellenstruktur

im Flanschbereich, was die Stabilisation des Flanschbereiches ermöglicht und somit einer

Faltenbildung entgegenwirkt. Neben ökonomischen und ökologischen Vorteilen ermöglicht

das Tiefziehen mit makrostrukturierten Werkzeugen die Steuerung des Materialflusses durch

Einstellung der Eintauchtiefe und reduziert das Rückfederungsverhalten des Bauteils. Durch

Tiefziehen mit makrostrukturiertem Werkzeug kann eine Vielzahl positiver Effekte erreicht

werden:

Reduzierung der Kontaktfläche

Reduzierung des Integrals der Flächenpressung über die Kontaktfläche

Reduzierung der gesamten Stempelkraft durch Minimierung der Reibkräfte

Zusammenfassung

96

Einsparung von Reinigungszeiten durch den Wegfall von Schmierstoffen und der

Entsorgungskosten

Erweiterung des Prozessfensters bezüglich der Bodenreißer durch Verringerung der

Stempelkraft

Erweiterung der Prozessstabilität bezüglich der Faltenbildung durch Ausnutzung des

Wechselbiegemechanismus

Kompensation des Rückfederungsverhaltens des Bauteils durch den erzeugten

Wechselbiegemechanismus in Form einer nicht reibungsbehafteten Rückhaltekraft

Möglichkeit zur Kontrolle der zusätzlichen isotropen und kinematischen Verfestigung

im Flanschbereich durch den Wechselbiegemechanismus

Möglichkeit zur Werkstoffflusssteuerung durch Einstellung der Eintauchtiefe

Keine Tuschierung bei neuen Werkzeugen erforderlich

Da die Prozessstabilität von den Prozessparametern, d. h. Eintauchtiefe und Wellenlänge,

abhängt, ist es für ein zeiteffektives Werkzeugdesign notwendig, die Prozessstabilität anhand

analytischer Modelle und Kriterien schnell und präzise vorhersagen zu können. Aus diesem

Grund wurde der entwickelte Tiefziehprozess analytisch mittels Energiemethode und auch

Beulanalyse modelliert, um das Auftreten von Bodenreißern und Faltenbildung vorhersagen

zu können. Die Machbarkeit des Prozesses sowie die Genauigkeit der analytischen

Berechnungen wurden durch numerische Simulationen und experimentelle Untersuchungen

überprüft. Die Ergebnisse zeigten eine hohe Übereinstimmung mit den numerischen und

experimentellen Ergebnissen.

Damit die Übertragbarkeit des hier vorgestellten Ansatzes auf kompliziertere Geometrien

gewährleistet werden kann, wurden im Rahmen dieser Arbeit sowohl makrostrukturierte als

auch konventionelle Werkzeuge zum Tiefziehen eines T-Napfes konstruiert und gefertigt. Die

beiden Kriterien zur Vorhersage der Prozessgrenzen wurden für die Werkzeugauslegung

angewandt. Darauf basierend wurden keine Bodenreißer und keine Faltenbildung erwartet,

was sich experimentell bestätigte. Darüber hinaus zeigten die Resultate, dass die maximale

Stempelkraft beim Tiefziehen mit makrostrukturiertem Werkzeug unter schmierstofffreien

Bedingungen nur marginal höher als jene beim konventionellen Tiefziehen mit Distanzring

unter Einsatz von Schmierstoff ist. Des Weiteren zeigte sich, dass das konventionelle

Tiefziehen ohne Einsatz von Schmierstoff durch die aufgebrachte Niederhaltekraft zum

Bodenreißer kam. Allerdings führte der Einsatz eines Distanzrings beim konventionellen

Tiefziehen unter schmierstofffreien Bedingungen ebenfalls zu einem stabilen Prozess ohne

wesentlichen Unterschied bezüglich der Stempelkraft gegenüber dem Tiefziehen mit

makrostrukturiertem Werkzeug. Jedoch treten bei letzterem Prozess durch seine Fähigkeit,

den Materialfluss während des Prozesses zu kontrollieren, keine Beule im Bodenbereich des

Bauteils auf.

Nachfolgend werden zwei weitere Ansätze erwähnt, um die Stempelkraft weiter zu

reduzieren. Der Einsatz von gelagerten Kugeln als Makrostruktur führt dazu, dass Rollreibung

statt Gleitreibung Anwendung findet, was die Reibkräfte weiter reduzieren und das

Zusammenfassung

97

Prozessfenster vergrößern kann. Auch wenn dies in der Anwendung und Umsetzung ein

komplizierter Ansatz seien mag, kann er doch als Vision für die zukünftige Forschung

verstanden werden. Außerdem zeigte sich, dass eine lokal angepasste Wellenlänge bezüglich

der tangentialen Spannung einen Beitrag zur weiteren Reduzierung der Stempelkraft leisten

kann.

Obwohl die Auslegung und Herstellung der makrostrukturierten Werkzeuge aufgrund ihrer

Form und zusätzlicher verschleißminimierender Maßnahmen, wie Beschichtung und

Oberflächenstrukturierung, relativ zeit- und kostenintensiv seien kann, bietet ihr Einsatz

zahlreiche ökonomische und ökologische Vorteile in der Fertigung. Da die Hauptaufgabe der

Makrostrukturierung eine Reduktion der Reibkräfte ist, kann sie auch in Gegenwart von

Schmierstoffen im Rahmen einer Minimalmengenschmierung zum Einsatz kommen. Das neu

entwickelte Tiefziehwerkzeug kann in Kombination mit herkömmlichen Schmiersystemen die

Reibung hochgradig verringern und somit zu einer signifikanten Vergrößerung des

Prozessfensters führen.

.

List of Figures

98

III. LIST OF FIGURES

Figure 1-1: Structure of Audi A4 limousine with hang on parts [2].......................................... 1

Figure 2-1: Deep drawing mechanism, geometrical variables, and acting stresses [10] ......... 4

Figure 2-2: Individual components of the total forming force in deep drawing of the rotationally

symmetric cup ....................................................................................................................... 5

Figure 2-3: Process window in deep drawing ........................................................................ 6

Figure 2-4: The Stribeck curve showing the onset of various lubrication mechanisms [29] .... 8

Figure 2-5: Comparison between bulk and sheet metal forming regarding the surface pressure

and the resulting surface enlargement according to [40] ...................................................... 10

Figure 2-6: An overview of COULOMB [37], shear friction [38], OROWAN [41], and SHAW [42]

friction models ..................................................................................................................... 11

Figure 2-7: Friction force as a function of normal force and the range of applications for various

manufacturing processes (the ranges shown are for unlubricated cases) according to [45] . 12

Figure 2-8: Threefold diagrams of amorphous carbon types and the properties of ta-C coating

[95] ...................................................................................................................................... 17

Figure 2-9: Average of specific roughness values for the substrate, coated and brushed

surfaces [98] ........................................................................................................................ 18

Figure 2-10: Comparison between the friction coefficient of the ta-C coated tool in lubricant-

free conditions and uncoated tools in lubricant-free and lubricated conditions resulting from

draw-bend tests with a range of results ............................................................................... 18

Figure 2-11: Different micro-structuring technologies differing from the surface fabrication

speed and structure size [106] ............................................................................................. 20

Figure 2-12: Direct Laser Interface Pattering; A) schematic of the interference principle, B) line-

type pattern with corresponding two laser beams and C) dot-type pattern with corresponding

three laser beams [111] ....................................................................................................... 20

Figure 2-13: Comparison between unstructured and DLIP structured ta-C coating regarding

wear reduction by the draw-bend test; A) relative wear and B) structure height differences

before and after testing [114] ............................................................................................... 21

List of Figures

99

Figure 2-14: Schematic overview of a draw-bend test for the calculation of the friction

coefficient under high non-uniform pressure ....................................................................... 23

Figure 3-1: Advantages of a lubricant-free deep drawing process according to [128] ........... 26

Figure 4-1: Reduction of punch force in deep drawing process through 50% reduction of A)

friction coefficient and B) blankholder force ......................................................................... 28

Figure 4-2: Surface pressure in A) conventional and B) macro-structured deep drawing tool 29

Figure 4-3: Tooling arrangement for macro-structured deep drawing; A) Tip to Tip and B) Tip

to Hutch ............................................................................................................................... 30

Figure 4-4: Comparison between the A) conventional and B) macro-structured deep drawing

process ................................................................................................................................ 31

Figure 4-5: Process window in A) a conventional deep drawing process, B) macro-structuring

the tools with Tip to Tip tooling arrangement, and C) macro-structuring with Tip to Hutch

tooling arrangement ............................................................................................................. 32

Figure 4-6: Geometrical parameters of induced alternating bending during the process ...... 32

Figure 4-7: Change of the alternating bending radius as a function of immersion depth and

wavelength; A) s0 = 0.6 mm and s0 = 0.8 mm ..................................................................... 33

Figure 4-8: Energy terms in the conventional and macro-structured deep drawing processes

............................................................................................................................................ 33

Figure 5-1: Hydraulic press machine BUP 600 ..................................................................... 35

Figure 5-2: RÖCHER RZP250 Press ..................................................................................... 36

Figure 5-3: Flow curve of workpiece materials; A) DC04 and B) AA5182 ............................. 40

Figure 5-4: Geometry and dimensions of A) conventional and B) macro-structured draw

bending of U-Channel forming tools .................................................................................... 41

Figure 5-5: Tools for deep drawing of axial symmetric parts; A) Conventional, rotationally

symmetric, B) macro-structured, rotationally symmetric and C) macro-structured, rectangular

............................................................................................................................................ 42

Figure 6-1: Comparison of time needed for tool design between two procedures ............... 43

List of Figures

100

Figure 6-2: Stress state on the flange area of the deep drawing process with macro-structured

tools. .................................................................................................................................... 44

Figure 6-3: Buckling of rectangular plate under uniaxial compression and tension ............... 45

Figure 6-4: Comparison between the calculated punch force from the method of Siebel, Bauer

and the principle of virtual work with experimental results according to [9] ......................... 46

Figure 6-5: Individual components of the total forming energy for deep drawing of a rotationally

symmetric cup with a macro-structured tool ........................................................................ 47

Figure 6-6: Principle of the analytical calculation of ideal forming energy in the flange area

according to the principle of virtual work [153] ..................................................................... 48

Figure 6-7: Principle of the analytical calculation of the bending energy, according to the

principle of virtual work [153] ............................................................................................... 49

Figure 6-8: Stress state in the flange area of the rectangular cup; A) Distribution of tangential

stress in the flange area according to [157] and B) the stress state in rotationally symmetric

and straight parts of the flange area ..................................................................................... 50

Figure 6-9: Free end part of the sheet metal; A) Controlling the stability of the process

regarding wrinkling and B) Progress of the tangential stress at the free end part of the sheet

during deep drawing ............................................................................................................ 54

Figure 6-10: Fracture locus in the space of the stress triaxiality and the equivalent plastic strain

to fracture ............................................................................................................................ 55

Figure 6-11: Influence of alternating bending on the kinematic hardening behaviour of the

workpiece ............................................................................................................................ 57

Figure 6-12: The ring splitting method to determine circumferential residual stress in deep

drawing cups; A) Cutting area in a rotationally symmetric deep drawing cup for the splitting

test, B) springback in the split ring and C) tangential residual stress at the inner and outer cup

surfaces ............................................................................................................................... 57

Figure 7-1: The required punch force for forming the U-Channel with conventional and macro-

structured tools in lubricant-free as well as lubricated conditions......................................... 62

Figure 7-2: Results of the numerical parameter analysis of the process window ................. 64

Figure 7-3: Geometry of the sheet metals used for the control of process stability ............. 64

List of Figures

101

Figure 7-4: Results of the experimental tests for the control of process stability regarding

wrinkling by deep drawing with macro-structured tools ....................................................... 65

Figure 7-5: Schematic Overview for springback geometry for a U-Channel [172] ................. 67

Figure 7-6: Springback behaviour of the workpieces with and without alternating bending for

A) DC04 and B) AA5182 [148] .............................................................................................. 68

Figure 7-7: Springback angle of U-Channels under different hardening types for A) DC04 and

B) AA5182 ........................................................................................................................... 69

Figure 7-8: Comparing the analytical calculated total forming energy with experimental and

FEM results for deep drawing of rotationally symmetric cups with outer radius of A) 90 mm

and B) 100 mm [130] ........................................................................................................... 70

Figure 7-9: Comparing the analytically calculated total forming energy with experimental and

FEM results for the deep drawing of rectangular cups ......................................................... 70

Figure 7-10: Prediction of wrinkling for samples with a sheet thickness of A) s0 = 0.6 mm, B)

s0 = 0.3 mm and C) s0 = 0.5 mm .......................................................................................... 72

Figure 7-11: Prediction of bottom cracks through the critical punch force for sheet metals with

outer radius of A) 90 mm and B) 100 mm ............................................................................ 73

Figure 7-12: Tangential residual stress over the sheet thickness in the cup made by

conventional and macro-structured processes from A) DC04 and B) AA5182. ..................... 74

Figure 7-13: Results of the ring splitting test for cups made by conventional and macro-

structured tools from A) DC04 and B) AA5182. .................................................................... 75

Figure 8-1: Detailed view and components of the T-Cup deep drawing tool ........................ 78

Figure 8-2: Determination of the optimum initial blank geometry using Slip-line field theory 79

Figure 8-3: The superposing method for analysing the deep drawing of the T-Cup .............. 80

Figure 8-4: Progress of the deep drawing of the T-Cup with macro-structured tools............ 81

Figure 8-5: Load-displacement curve for the deep drawing of the T-Cup ............................. 82

Figure 8-6: Comparison between the maximum punch force from the experimental tests, FEM

calculations and the analytical model ................................................................................... 82

List of Figures

102

Figure 8-7: Buckling analysis to predict the process stability regarding wrinkling for the deep

drawing of the T-Cup ........................................................................................................... 83

Figure 8-8: Comparison between the conventional and macro-structured deep drawing

processes; A) Difference between the conventional and macro-structured deep drawing

processes regarding the total punch force and B) the progress of the blankholder force as a

function of drawing depth .................................................................................................... 84

Figure 8-9: Comparison between the part properties made by conventional and macro-

structured tools in lubricant-free conditions; A) Conventional tool and B) macro-structured . 85

Figure 9-1: Using rotary elements as macro-structured elements; A) Schematic of conventional

and macro-structured deep drawing tools, B) comparison between the conventional and

macro-structured deep drawing process regarding the punch force [114] ............................ 88

Figure 9-2: Macro-structured tool with variable wavelength; A) Increasing the amount of

tangential compressive stress as a result of strain hardening, and B) developing a concept for

a macro-structured tool with variable wavelength ................................................................ 89

Figure 9-3: Macro-structured deep drawing tool with variable wavelength .......................... 90

Figure 9-4: Comparison between macro-structured tools with constant and variable

wavelengths regarding the punch force and enlargement of the process window for sheet

metals with outer radius of A) 90 mm and B) 100 mm ........................................................ 91

Figure 10-1: Procedural methods for realisation of the lubricant-free deep drawing process 92

List of Tables

103

IV. LIST OF TABLES

Table 2-1: Comparison between different forming process regarding their economic

importance [8] ........................................................................................................................ 3

Table 2-2: Common equations used to calculate the friction coefficient for draw-bend test. 24

Table 5-1: Chemical composition of the studied tool steel (in wt. %) [132] .......................... 37

Table 5-2: Low-carbon steel grades with mechanical properties and chemical components

[134] .................................................................................................................................... 38

Table 5-3: Mechanical properties of Al 5182 [140] ............................................................... 39

Table 5-4: Chemical components of Al 5182 in wt% [117] ................................................... 39

Table 5-5: Parameters of VOCE model for both testing materials .......................................... 40

Table 7-1: Bauschinger coefficient of testing materials [165]. .............................................. 67

Table 7-2: Opening gap of split rings based on experimental tests and calculated values. ... 75

Table 8-1: Surface-related properties of modular deep drawing tool .................................... 77

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